A Tutorial on Euler Angles and Quaternions
|
|
- Brian Britton Marsh
- 5 years ago
- Views:
Transcription
1 A Tutorial on Euler Angles and Quaternions Moti Ben-Ari Department of Science Teaching Weimann Institute of Science Version.0.1 c b Moti Ben-Ari. This work is licensed under the Creative Commons Attribution-ShareAlike 3.0 Unported License. To view a cop of this license, visit b-sa/3.0/ or send a letter to Creative Commons, Castro Street, Suite 900, Mountain View, California, 901, USA.
2 Chapter 1 Introduction You sitting in an airplane at night, watching a movie displaed on the screen attached to the seat in front of ou. The airplane gentl banks to the left. You ma feel the slight acceleration, but ou won t see an change in the position of the movie screen. Both ou and the screen are in the same frame of reference, so unless ou stand up or make another move, the position and orientation of the screen relative to our position and orientation won t change. The same is not true with respect to our position and orientation relative to the frame of reference of the earth. The airplane is moving forward at a ver high speed and the bank changes our orientation with respect to the earth. The transformation of coordinates (position and orientation) from one frame of reference is a fundamental operation in several areas: flight control of aircraft and rockets, movement of manipulators in robotics, and computer graphics. This tutorial introduces the mathematics of rotations using two formalisms: (1) Euler angles are the angles of rotation of a three-dimensional coordinate frame. A rotation of Euler angles is represented as a matri of trigonometric functions of the angles. () Quaternions are an algebraic structure that etends the familiar concept of comple numbers. While quaternions are much less intuitive than angles, rotations defined b quaternions can be computed more efficientl and with more stabilit, and therefore are widel used. The tutorial assumes an elementar knowledge of trigonometr and matrices. The computations will be given in great detail for two reasons. First, so that ou can be convinced of the correctness of the formulas, and, second, so that ou can learn how to do them ourselves, in case ou come across a contet that uses different definitions or notation. For a comprehensive presentation of quaternions using vector algebra, see: John Vince. Quaternions for Computer Graphics, 011, Springer. 1
3 Chapter Rotations in two dimensions We start b discussing rotations in two dimensions; the concepts and formulas should be familiar and a review will facilitate learning about rotations in three dimensions..1 Cartesian and polar coordinates A vector or a position (the tip of the vector) in a two-dimensional space can be given either in cartesian coordinates (, ) or in polar coordinates (r, φ), relative to a frame of reference: φ r The formulas for transforming one representation to another are: r cos φ r sin φ r + φ tan 1.. Rotating a vector Suppose now that the vector is rotated b the angle θ: r θ φ r
4 The new vector s polar coordinates are (r, φ + θ) and its cartesian coordinates can be computed using the trigonometric identities for the sum of two angles: r cos(φ + θ) r(cos φ cos θ sin φ sin θ) (r cos φ) cos θ (r sin φ) sin θ cos θ sin θ, r sin(φ + θ) r(sin φ cos θ + cos φ sin θ) (r sin φ) cos θ + (r cos φ) sin θ cos θ + sin θ sin θ + cos θ. The transformation of the position (, ) to (, ) caused b a rotation through the angle θ can be epressed in matri notation as: Eample [ [ cos θ sin θ Let r 1 and let both θ and φ be 30 so that: sin θ cos θ [ r cos φ cos φ cos θ cos 30 3/ r sin φ sin φ sin θ sin 30 1/. We can now compute the values of and : [ [ [ [ 3/ 1/ 3/ 1/ 3/ 1/ The result makes sense since φ + θ 60. ( 3/) (1/) 1/ 3/ + 1/ 3/. [ 1/. 3/.3 A geometric derivation of the rotation matri The rotation matri can be derived geometricall. Rather than look at the vector, let us look at its and components and rotate them (counterclockwise) b θ (Figure.1). The - and - components are rotated b the angle θ so that the OAB becomes OA B. Construct two right triangles: (1) Drop the perpendicular from A to the -ais to form the right triangle OA C; (b) Construct a line through A parallel to the -ais and a line through B parallel to the -ais. The intersect at C to form the right triangle B A C. Since A C is parallel to CO, C A O A OC θ. OA B is a right angle and A B C is a right triangle so A B C θ. The sides of the two right triangles are labeled with 3
5 B (, ) θ cos θ B (, ) C sin θ 90 θ A sin θ O θ cos θ C A Figure.1: Geometric derivation of the rotation matri their lengths obtained b trigonometr. It is now eas to see that is the length of OC minus the length of A C and that is the length of CA plus the length of C B : cos θ sin θ sin θ + cos θ.. Rotating the frame of reference The effect of rotating a vector b θ is the same as the effect obtained b rotation of the frame of reference. The left diagram below shows a vector in the frame of reference A. The right diagram shows the same vector after frame A is rotated θ relative to a new frame B. ( A, A ), ( B, B ) A r ( A, A ) A B r A φ A φ θ B Given the coordinates ( A, A ) of the points in the frame A: A r cos φ A r sin φ,
6 what are its coordinates ( B, B ) in the frame B? The geometr of this diagram is the same as the geometr of the diagram in Section., so we can use the same equation: Eample [ B B [ cos θ sin θ sin θ cos θ [ A A. (.1) Consider the point ( A, A ) ( 3/, 1/) in frame A. What are the coordinates of the point in frame B? As before, the computation gives: [ B B [ 3/ 1/ [ 1/ 3/ 3/ 1/ [ 1/. 3/.5 Multiple rotations Rotate the coordinate frame B b ψ relative to a new frame C. ( A, A ), ( B, B ), ( C, C ) A B C r B A φ θ ψ C If we know ( B, B ), we can multipl it b the rotation matri for ψ to obtain ( C, C ). Let us denote the rotation matri from frame A to frame B b R B A and the rotation matri from frame B to frame C b R C B. Given pa, the coordinates of a point in frame A, its coordinates in frame B are p B R B A pa, and p C, its coordinates in frame C, are p C R C B pb. Combining the two equations we have: p C R C B pb R C B RB A pa. Let R C A RC B RB A. This is the rotation matri from A to C, so we can obtain the coordinates p C directl from p A b: p C R C A pa. Eample Let ψ 30 so R C B RB A. Then: [ C C [ 3/ 1/ [ [ 1/ 1/ 0. 3/ 3/ 1 5
7 From the drawing we see the vector lies on the C -ais which corresponds to the coordinates (0, 1). Let us now compute: R C A RC B RB A [ 3/ 1/ [ 1/ 3/ 3/ 1/ [ 1/ 1/ 3/. 3/ 3/ 1/ The computation p C R C A pa is: [ [ C 1/ [ 3/ 3/ 3/ 1/ 1/ C [ 0 1, the same result obtained in the two-stage computation. Eample Consider the vector (1, 0) ling on the -ais of frame A. Rotate A b 15 to frame B and then rotate frame B b 30 to frame C. Hopefull, the coordinates of the vector in frame C will be ( /, /), because the vector makes an angle of 5 with the -ais of frame C. The values of the trigonometric functions for 15 are: cos 15, sin 15. First rotate b 15 : [ B Now rotate b 30 : [ C C B [ 3 1 [ [ 6+ 6 If we multipl the two rotation matrices: [ [ 3 1 [ the result is the rotation matri for a rotation b 5. [ [ [ [ If we multipl the two matrices in the opposite order, we get the same result: [ [ [ This shows that rotating b 15 and then b 30 gives the same result as rotating b 30 and then b 15. This is not surprising because we are familiar with the commutativit of turning knobs clockwise and counterclockwise. Unfortunatel, we will not be so luck when we discuss three-dimensional rotations. 6,.
8 Chapter 3 Rotations in three dimensions The concept of coordinate transformations in three-dimensions is the same as in two dimensions, however, the mathematics are more complicated. Furthermore, man of us find it difficult to visualie eplanations of three-dimensional motion when all we are shown is two-dimensional representations of three-dimensional objects. The great eighteenth-centur mathematician Leonhard Euler proved that an arbitrar three-dimensional rotation can be obtained b three individual rotations around the aes. In his honor, the angles of the rotations are called Euler angles. 3.1 Rotations around the three aes A two-dimensional - coordinate frame can be considered to be part of the threedimensional coordinate frame b adding a -ais perpendicular to the - and -aes. Here is a two-dimensional representation of the three-dimensional coordinate frame: up in left right out down The -ais is drawn left and right on the paper and the -ais is drawn up and down on the paper. The diagonal line represents the -ais, which is perpendicular to the other two aes and therefore its positive direction is out of the paper and its negative direction is into the paper. The standard -- coordinate frame has the positive directions of its aes pointing in the directions (right, up, out) as shown in the left diagram: 7
9 Rotate the coordinate frame counterclockwise around the -ais, so the -ais which remains unchanged (right diagram). The new orientation of the frame is (up, left, out). Consider now a rotation of 90 around the -ais: This causes the -ais to jump out of the page and the -ais to fall down onto the page, resulting in the orientation (right, out, down). Finall, consider a rotation of 90 around the -ais (left diagram): The -ais drops below the paper and the -ais falls right onto the paper. The new position of the frame is (in, up, right). 3. The right hand rule The -ais must be perpendicular to the - and -aes, but there are two orientations for this line: one in which the positive direction of the ais is out of the paper and another where the positive direction of the ais is into the paper. It doesn t matter which one we choose as long as the choice is consistent. The conventional choice is the right hand rule. Curl the fingers of our right hand so that the go b the shortest path from the one ais to another ais. Your thumb now points in the positive direction of the third ais. For the familiar - and -aes on paper, curl our fingers on the short 90 path from the -ais to the -ais (not on the long 70 path from the -ais to the -ais). When ou do so our thumb points out of the paper and this is taken as the positive direction of the -ais. The following diagram shows the right-hand coordinate sstem displaed with each of the three aes pointing out of the paper: 8
10 The three rotations are: Rotate from to around, Rotate from to around, Rotate from to around. 3.3 Matrices for three-dimensional rotations A three-dimensional rotation matri will be a 3 3 matri because each point p in a frame has three coordinates p, p, p that must be moved. Consider first a rotation around the -ais. The and coordinates are rotated as in two dimensions and the coordinate remains unchanged. Therefore, the matri is: cos θ sin θ 0 sin θ cos θ For a rotation about the -ais, the coordinate is unchanged and the and coordinates are transformed as if the were the and coordinates of a rotation around the -ais: cos θ sin θ. 0 sin θ cos θ For a rotation about the -ais, the coordinate is unchanged and the and coordinates are transformed as if the were the and coordinates of a rotation around the -ais: cos θ 0 sin θ sin θ 0 cos θ It ma seem strange that the signs of sin θ have changed. To convince ourself that this matri is correct, redraw the diagram in Section. and perform the computations, substituting for and for.. 9
11 3. Multiple rotations Here is an eample of three successive rotations of a coordiante frame: first, 90 around the -ais, then 90 around the -ais and finall 90 around the -ais: This is called a Euler angle rotation. The final orientation is (in, up, right). There is one caveat to composing rotations: three-dimensional rotations do not commute. Let us demonstrate this b a simple sequence of two rotations. Consider a rotation of 90 around the -ais, followed b a rotation of 90 around the -ais: The result can be epressed as (up, out, right). Now consider the commuted operation: a rotation of 90 around the -ais, followed b a rotation of 90 around the -ais: The result can be epressed as (out, left, down) which is not the same as the previous orientation. There are three aes so there should be sequences of Euler angles. However, there is no point in rotating around the same ais twice in succession because the same result can be obtained b rotating once b the sum of the angles, so there are onl 3 1 different Euler angle sequences:. 10
12 3.5 Euler angles for transforming a vector Let us return to the sequence of Euler angle rotations at the beginning of Section 3.. Consider the vector (or point) (1, 1, 1) in the final frame: (1, 1, 1) What are its coordinates in the previous frame that was rotated around the -ais to become this frame? Rotating the previous frame (not the vector), we find that the point s coordinates are (1, 1, 1): (1, 1, 1) What are its coordinates in the previous frame that was rotated around the -ais to become this frame? Rotating the previous frame around the -ais gives (1, 1, 1): (1, 1, 1) Finall, what are its coordinates in the previous frame that was rotated around the -ais to become this frame? Rotating the previous frame around the -ais gives the coordinates in the original frame (1, 1, 1): (1, 1, 1) These coordinates can be computed from the rotation matrices for the rotations around the three aes. First around the -ais: ,
13 then around the -ais: and finall around the -ais: ,. 3.6 One matri for three rotations Instead of performing the three multiplications of a vector b a matri, we can multipl the matrices for the three Euler angles together. The result is the matri for the complete rotation: The vector (1, 1, 1) can now be multiplied b this matri and we obtain the epected value: The matri for an arbitrar rotation The previous section showed an eample of a matri that rotates a vector around the aes b 90 each. The matri for arbitrar rotations around these aes is obtained b multipling the matrices for each ais using arbitrar angles: a rotation of ψ around the -ais, a rotation of θ around the ais and a rotation of φ around the -ais. The resulting matri is computed as follows. First multipl the rotation around the -ais b the rotation around the -ais: R (θ) R (φ) cos θ 0 sin θ sin θ 0 cos θ cos φ sin φ 0 sin φ cos φ cos θ sin θ sin φ sin θ cos φ 0 cos φ sin φ sin θ Then, multipl the result b the rotation around the -ais: R (ψ)(θ)(φ) R (ψ) (R (θ) R (φ) ) cos ψ sin ψ 0 sin ψ cos ψ cos θ sin φ cos θ cos φ cos θ sin θ sin φ sin θ cos φ 0 cos φ sin φ sin θ cos θ sin φ cos θ cos φ..
14 R (ψ)(θ)(φ) R (ψ) (R (θ) R (φ) ) cos ψ cos θ cos ψ sin θ sin φ cos ψ sin θ cos φ+ sin ψ cos φ sin ψ sin φ sin ψ cos θ sin ψ sin θ sin φ+ sin ψ sin θ cos φ cos ψ cos φ cos ψ sin φ sin θ cos θ sin φ cos θ cos φ. For the rotation R (90)(90)(90), we can evaluate the matri to obtain the matri we computed from the individual rotations:
15 Chapter Quaternions.1 Comple numbers A comple number is a pair written formall as a + bi, where a, b are real numbers and i is a new smbol. Arithmetic is performed as if comple numbers were formal polnomials with indeterminate i and then simplified using the identif i 1: (a + bi) + (c + di) (a + c) + (b + d)i (a + bi) (c + di) ac + adi + bci + bdi (ac bd) + (ad + bc)i. Comple numbers can represent vectors and points in the plane. The number + i is identified with the vector from (0, 0) to the point (, ). Comple multiplication can be used to rotate a vector. B Euler s formula: so: e iθ (cos θ + i sin θ), e iθ ( + i) (cos θ + i sin θ)( + i) ( cos θ sin θ) + i( sin θ + cos θ), which is the two-dimensional rotation of (, ) b θ: [ [ cos θ sin θ sin θ cos θ [. In 183, the mathematician William R. Hamilton discovered that three-dimensional rotations can be described b a generaliation of comple numbers called quaternions.. The algebra of quaternions A quaternion is a -tuple written formall as q 0 + q 1 i + q j + q 3 k, where q i are real numbers and the smbols i, j, k satisf the following identities: i j k 1 ij k, ji k jk i, kj i ki j, ik j. Multiplication of these smbols i, j, k are not commutative. To remember the sign of a multiplication, consider the smbols to be arranged cclicall: 1
16 i k j If two smbols to be multiplied are adjacent clockwise, the sign is plus, whereas if the are adjacent counterclockwise, the sign is minus. Arithmetic is performed formall and the results are simplified using the identities. Here are the formulas for adding and multipling two quaternions: (q 0 + q 1 i + q j + q 3 k) + (p 0 + p 1 i + p j + p 3 k) (q 0 + p 0 ) + (q 1 + p 1 )i + (q + p )j + (q 3 + p 3 )k (q 0 + q 1 i + q j + q 3 k) (p 0 + p 1 i + p j + p 3 k) (q 0 p 0 + q 1 p 1 i + q p j + q 3 p 3 k ) + (q 0 p 1 i + q 1 p 0 i + q p 3 jk + q 3 p kj) + (q 0 p j + q p 0 j + q 1 p 3 ik + q 3 p 1 ki) + (q 0 p 3 k + q 3 p 0 k + q 1 p ij + q p 1 ji) (q 0 p 0 q 1 p 1 q p q 3 p 3 ) + (q 0 p 1 + q 1 p 0 + q p 3 q 3 p )i + (q 0 p + q p 0 q 1 p 3 + q 3 p 1 )j + (q 0 p 3 + q 3 p 0 + q 1 p q p 1 )k..3 The conjugate and absolute value of a comple number Comple numbers form a field, because multiplication is commutative and ever nonero comple number has an inverse: (a + bi)(a bi) a b i a + b (a + bi)(a bi) a + b 1 (a + bi) 1 a bi a + b. For a comple number a + bi, define its conjugate as a bi and its norm as a + b. Then its inverse can be written as: 1.. The conjugate and norm of a quaternion The corresponding definitions for quaternions are as follows. 15
17 Given q q 0 + q 1 i + q j + q 3 k, its conjugate is: and its norm is: q q 0 q 1 i q j q 3 k, q Ever nonero quaternion has an inverse: q 0 + q 1 + q + q 3. q q (q 0 + q 1 i + q j + q 3 k)(q 0 q 1 i q j q 3 k) (q 0 q 0 q 1 q 1 i q q j q 3 q 3 k ) + ( q 0 q 1 + q 1 q 0 q q 3 + q 3 q )i + ( q 0 q + q q 0 + q 1 q 3 q 3 q 1 )j + ( q 0 q 3 + q 3 q 0 q 1 q + q q 1 )k q 0 + q 1 + q + q 3. q 1 q q. Quaternions do not form a field because multiplication is not commutative..5 Quaternions for rotations A vector in three-dimensional space can be epressed as a pure quaternion, a quaternion with no real part: q 0 + i + j + k. A rotation is epressed b a quaternion q R with the additional requirement that its norm q R be equal to 1. A rotation from one coordinate frame A to another B is given b the conjugation operation: q B q R q A q R. This operation will result in a quaternion q B which is also a vector since it has no real part: q R q A q R (q 0 + q 1 i + q j + q 3 k)(i + j + k)(q 0 q 1 i q j q 3 k) ((q 0 + q 1 q q 3) + (q 1 q q 0 q 3 ) + (q 0 q + q 1 q 3 ))i + ((q 0 q 3 + q 1 q ) + (q 0 q 1 + q q 3) + (q q 3 q 0 q 1 ))j + ((q 1 q 3 q 0 q ) + (q 0 q 1 + q q 3 ) + (q 0 q 1 q + q 3))k. The detailed multiplication is given in Appendi A. This computation does not require the computation of trigonometric functions, onl the addition and multiplication of real numbers..6 From a quaternion to a rotation matri We need to compute the quaternion of a rotation from Euler angles and rotation matrices and conversel from quaternions back to angles and matrices. 16
18 Epressing the formula for q R q A q R as a matri gives: M q 0 + q 1 q q 3 (q 1 q q 0 q 3 ) (q 0 q + q 1 q 3 ) (q 0 q 3 + q 1 q ) q 0 q 1 + q q 3 (q q 3 q 0 q 1 ). (q 1 q 3 q 0 q ) (q 0 q 1 + q q 3 ) q 0 q 1 q + q 3 The matri can be simplified since the norm of a rotation quaternion is 1: q q 0 + q 1 + q + q 3 1, from which we have the following equations: q q 3 q 0 + q 1 1 q 1 q 3 q 0 + q 1 q 1 q q 0 + q 3 1. After simplification the matri becomes: q 0 + q q 1q q 0 q 3 q 0 q + q 1 q 3 M q 0 q 3 + q 1 q q 0 + q 0.5 q q 3 q 0 q 1. q 1 q 3 q 0 q q 0 q 1 + q q 3 q 0 + q From a matri of angles to a quaternion Given a rotation matri M, we can compute the quaternion that represents the same rotation. First compute the trace, the sum of the elements on the main diagonal of the matri: Trace(M) M 11 + M + M 33 (3q 0 + q 1 + q + q 3 1.5) (3q 0 + (1 q 0) 1.5) (q 0 0.5) q 0 1. We can solve this equation for q 0 : Trace(M) + 1 q 0. Once we have q 0, we can obtain q 1 from M 11 : ( Trace(M) + 1 M 11 (q 0 + q 1 0.5) q 1 M Trace(M) q ).
19 Similarl, q and q 3 can be computed from M and M 33 : q q 3 M M Trace(M). + 1 Trace(M)..8 From Euler angles to a quaternion Given a rotation b an angle, the corresponding quaternion can be computed. Consider the rotation ψ around the -ais: M ψ cos ψ sin ψ 0 sin ψ cos ψ From the formulas in the previous section and using that Trace(M) cos ψ + 1: q 0 cos ψ cos ψ cos ψ, q 1 q cos ψ + 1 ( cos ψ + 1) 0, q ( cos ψ + 1) 1 cos ψ + sin ψ. Therefore: q ψ cos ψ + k sin ψ. Similarl, we can compute the quaternions corresponding to rotations around the - and -aes: q θ cos θ + j sin θ, q φ cos φ + i sin φ..9 Eample Consider a rotation of +90 around the -ais. It takes the point (1, 0, 0) to (0, 1, 0). The rotation matri is: cos 90 sin sin 90 cos
20 Using the formulas in Section.7 and computing Trace(M) , we have: q 0 q 1 q q 3 Trace(M) M Trace(M) M + 1 Trace(M) M Trace(M) Therefore: q R k. We can obtain the same result directl from the formula in Section.8:. q R cos 5 + k sin 5 + k. The point (1, 0, 0) is 0 + i + 0j + 0k as a quaternion so we compute the rotation as: q R i q R ( + k) i ( k) ( i + ki) ( k) ( i + j) ( k) 1 (i + j ik jk) 1 (i + j + j i) j. This is the point (0, 1, 0) as epected. 19
21 Appendi A Multiplication of three quaternions We compute the multiplication of quarternions to change a coordinate frame: q R q A q R (q 0 + q 1 i + q j + q 3 k)(i + j + k)(q 0 q 1 i q j q 3 k). Multipl the first two factors: (q 0 + q 1 i + q j + q 3 k)(i + j + k) q 0 i + q 0 j + q 0 k + q 1 i + q 1 ij + q 1 ik + q ji + q j + q jk + q 3 ki + q 3 kj + q 3 k q 1 q q 3 + q 0 i + q i q 3 i + q 0 j q 1 j + q 3 j + q 0 k + q 1 k q k. Now mulitpl on the right b (q 0 q 1 i q j q 3 k). The real part Multipl the first row b q 0, the second b q 1 i, the third b q j and the fourth b q 3 k: The coefficient of i q 0 q 1 q 0 q q 0 q 3 +q 0 q 1 + q 1 q q 1 q 3 +q 0 q q 1 q + q q 3 +q 0 q 3 + q 1 q 3 q q 3 0. Multipl the second row b q 0 the first b q 1 i, the fourth b q j (since kj i) and the third b q 3 k (since jk i): +q 0 q 0 + q 0 q q 0 q 3 +q 1 q 1 + q 1 q + q 1 q 3 +q 0 q + q 1 q q q q 0 q 3 + q 1 q 3 q 3 q 3. Collecting terms gives: (q 0 + q 1 q q 3 )+ (q 1 q q 0 q 3 )+ (q 0 q + q 1 q 3 ). 0
22 The coefficient of j Multipl the third row b q 0, the fourth b q 1 i (since ki j), the first b q j and the second b q 3 k (since ik j): +q 0 q 0 q 0 q 1 + q 0 q 3 q 0 q 1 q 1 q 1 + q 1 q +q 1 q + q q + q q 3 +q 0 q 3 + q q 3 q 3 q 3. Collecting terms gives: (q 1 q + q 0 q 3 )+ (q 0 q 1 + q q 3 )+ (q q 3 q 0 q 1 ). The coefficient of k Multipl the fourth row b q 0, the third b q 1 i (since ji k), the second b q j (since ij k and the first b q 3 k: +q 0 q 0 + q 0 q 1 q 0 q +q 0 q 1 q 1 q 1 + q 1 q 3 q 0 q q q + q q 3 +q 1 q 3 + q q 3 + q 3 q 3. Collecting terms gives: (q 1 q 3 q 0 q )+ (q q 3 + q 0 q 1 )+ (q 0 q 1 q + q 3 ). Collect the results in one quaternion ((q 0 + q 1 q q 3 ) + (q 1q q 0 q 3 ) + (q 0 q + q 1 q 3 ))i+ ((q 0 q 3 + q 1 q ) + (q 0 q 1 + q q 3 ) + (q q 3 q 0 q 1 ))j+ ((q 1 q 3 q 0 q ) + (q 0 q 1 + q q 3 ) + (q 0 q 1 q + q 3 ))k. 1
9.1 VECTORS. A Geometric View of Vectors LEARNING OBJECTIVES. = a, b
vectors and POLAR COORDINATES LEARNING OBJECTIVES In this section, ou will: View vectors geometricall. Find magnitude and direction. Perform vector addition and scalar multiplication. Find the component
More information9.1 VECTORS. A Geometric View of Vectors LEARNING OBJECTIVES. = a, b
vectors and POLAR COORDINATES LEARNING OBJECTIVES In this section, ou will: View vectors geometricall. Find magnitude and direction. Perform vector addition and scalar multiplication. Find the component
More informationEigenvectors and Eigenvalues 1
Ma 2015 page 1 Eigenvectors and Eigenvalues 1 In this handout, we will eplore eigenvectors and eigenvalues. We will begin with an eploration, then provide some direct eplanation and worked eamples, and
More informationGZW. How can you find exact trigonometric ratios?
4. Special Angles Aircraft pilots often cannot see other nearb planes because of clouds, fog, or visual obstructions. Air Traffic Control uses software to track the location of aircraft to ensure that
More informationAnalytic Geometry in Three Dimensions
Analtic Geometr in Three Dimensions. The Three-Dimensional Coordinate Sstem. Vectors in Space. The Cross Product of Two Vectors. Lines and Planes in Space The three-dimensional coordinate sstem is used
More informationCHAPTER 1 MEASUREMENTS AND VECTORS
CHPTER 1 MESUREMENTS ND VECTORS 1 CHPTER 1 MESUREMENTS ND VECTORS 1.1 UNITS ND STNDRDS n phsical quantit must have, besides its numerical value, a standard unit. It will be meaningless to sa that the distance
More informationECS 178 Course Notes QUATERNIONS
ECS 178 Course Notes QUATERNIONS Kenneth I. Joy Institute for Data Analysis and Visualization Department of Computer Science University of California, Davis Overview The quaternion number system was discovered
More informationPhysically Based Rendering ( ) Geometry and Transformations
Phsicall Based Rendering (6.657) Geometr and Transformations 3D Point Specifies a location Origin 3D Point Specifies a location Represented b three coordinates Infinitel small class Point3D { public: Coordinate
More information9.2. Cartesian Components of Vectors. Introduction. Prerequisites. Learning Outcomes
Cartesian Components of Vectors 9.2 Introduction It is useful to be able to describe vectors with reference to specific coordinate sstems, such as the Cartesian coordinate sstem. So, in this Section, we
More informationVectors Primer. M.C. Simani. July 7, 2007
Vectors Primer M.. Simani Jul 7, 2007 This note gives a short introduction to the concept of vector and summarizes the basic properties of vectors. Reference textbook: Universit Phsics, Young and Freedman,
More informationVectors in Two Dimensions
Vectors in Two Dimensions Introduction In engineering, phsics, and mathematics, vectors are a mathematical or graphical representation of a phsical quantit that has a magnitude as well as a direction.
More informationMath 123 Summary of Important Algebra & Trigonometry Concepts Chapter 1 & Appendix D, Stewart, Calculus Early Transcendentals
Math Summar of Important Algebra & Trigonometr Concepts Chapter & Appendi D, Stewart, Calculus Earl Transcendentals Function a rule that assigns to each element in a set D eactl one element, called f (
More informationTrigonometric. equations. Topic: Periodic functions and applications. Simple trigonometric. equations. Equations using radians Further trigonometric
Trigonometric equations 6 sllabusref eferenceence Topic: Periodic functions and applications In this cha 6A 6B 6C 6D 6E chapter Simple trigonometric equations Equations using radians Further trigonometric
More informationTwo conventions for coordinate systems. Left-Hand vs Right-Hand. x z. Which is which?
walters@buffalo.edu CSE 480/580 Lecture 2 Slide 3-D Transformations 3-D space Two conventions for coordinate sstems Left-Hand vs Right-Hand (Thumb is the ais, inde is the ais) Which is which? Most graphics
More informationCourse MA2C02, Hilary Term 2010 Section 4: Vectors and Quaternions
Course MA2C02, Hilary Term 2010 Section 4: Vectors and Quaternions David R. Wilkins Copyright c David R. Wilkins 2000 2010 Contents 4 Vectors and Quaternions 47 4.1 Vectors...............................
More informationVECTORS. 3-1 What is Physics? 3-2 Vectors and Scalars CHAPTER
CHAPTER 3 VECTORS 3-1 What is Physics? Physics deals with a great many quantities that have both size and direction, and it needs a special mathematical language the language of vectors to describe those
More informationAPPENDIXES. B Coordinate Geometry and Lines C. D Trigonometry E F. G The Logarithm Defined as an Integral H Complex Numbers I
APPENDIXES A Numbers, Inequalities, and Absolute Values B Coordinate Geometr and Lines C Graphs of Second-Degree Equations D Trigonometr E F Sigma Notation Proofs of Theorems G The Logarithm Defined as
More informationLesson #33 Solving Incomplete Quadratics
Lesson # Solving Incomplete Quadratics A.A.4 Know and apply the technique of completing the square ~ 1 ~ We can also set up any quadratic to solve it in this way by completing the square, the technique
More informationRe(z) = a, For example, 3 + 2i = = 13. The complex conjugate (or simply conjugate") of z = a + bi is the number z defined by
F COMPLEX NUMBERS In this appendi, we review the basic properties of comple numbers. A comple number is a number z of the form z = a + bi where a,b are real numbers and i represents a number whose square
More information8. BOOLEAN ALGEBRAS x x
8. BOOLEAN ALGEBRAS 8.1. Definition of a Boolean Algebra There are man sstems of interest to computing scientists that have a common underling structure. It makes sense to describe such a mathematical
More informationVectors and the Geometry of Space
Chapter 12 Vectors and the Geometr of Space Comments. What does multivariable mean in the name Multivariable Calculus? It means we stud functions that involve more than one variable in either the input
More informationwe must pay attention to the role of the coordinate system w.r.t. which we perform a tform
linear SO... we will want to represent the geometr of points in space we will often want to perform (rigid) transformations to these objects to position them translate rotate or move them in an animation
More informationName: Richard Montgomery High School Department of Mathematics. Summer Math Packet. for students entering. Algebra 2/Trig*
Name: Richard Montgomer High School Department of Mathematics Summer Math Packet for students entering Algebra 2/Trig* For the following courses: AAF, Honors Algebra 2, Algebra 2 (Please go the RM website
More informationGet Solution of These Packages & Learn by Video Tutorials on Matrices
FEE Download Stud Package from website: wwwtekoclassescom & wwwmathsbsuhagcom Get Solution of These Packages & Learn b Video Tutorials on wwwmathsbsuhagcom Matrices An rectangular arrangement of numbers
More informationVectors a vector is a quantity that has both a magnitude (size) and a direction
Vectors In physics, a vector is a quantity that has both a magnitude (size) and a direction. Familiar examples of vectors include velocity, force, and electric field. For any applications beyond one dimension,
More informationA.5. Complex Numbers AP-12. The Development of the Real Numbers
AP- A.5 Comple Numbers Comple numbers are epressions of the form a + ib, where a and b are real numbers and i is a smbol for -. Unfortunatel, the words real and imaginar have connotations that somehow
More informationHigher. Integration 1
Higher Mathematics Contents Indefinite Integrals RC Preparing to Integrate RC Differential Equations A Definite Integrals RC 7 Geometric Interpretation of A 8 Areas between Curves A 7 Integrating along
More informationSTUDY KNOWHOW PROGRAM STUDY AND LEARNING CENTRE. Functions & Graphs
STUDY KNOWHOW PROGRAM STUDY AND LEARNING CENTRE Functions & Graphs Contents Functions and Relations... 1 Interval Notation... 3 Graphs: Linear Functions... 5 Lines and Gradients... 7 Graphs: Quadratic
More informationwe must pay attention to the role of the coordinate system w.r.t. which we perform a tform
linear SO... we will want to represent the geometr of points in space we will often want to perform (rigid) transformations to these objects to position them translate rotate or move them in an animation
More informationPhys 221. Chapter 3. Vectors A. Dzyubenko Brooks/Cole
Phs 221 Chapter 3 Vectors adzubenko@csub.edu http://www.csub.edu/~adzubenko 2014. Dzubenko 2014 rooks/cole 1 Coordinate Sstems Used to describe the position of a point in space Coordinate sstem consists
More informationGround Rules. PC1221 Fundamentals of Physics I. Coordinate Systems. Cartesian Coordinate System. Lectures 5 and 6 Vectors.
PC1221 Fundamentals of Phsics I Lectures 5 and 6 Vectors Dr Ta Seng Chuan 1 Ground ules Switch off our handphone and pager Switch off our laptop computer and keep it No talking while lecture is going on
More informationMA123, Chapter 1: Equations, functions and graphs (pp. 1-15)
MA123, Chapter 1: Equations, functions and graphs (pp. 1-15) Date: Chapter Goals: Identif solutions to an equation. Solve an equation for one variable in terms of another. What is a function? Understand
More informationCumulative Review of Vectors
Cumulative Review of Vectors 1. For the vectors a! 1, 1, and b! 1, 4, 1, determine the following: a. the angle between the two vectors! the scalar and vector projections of a! on the scalar and vector
More information15. Eigenvalues, Eigenvectors
5 Eigenvalues, Eigenvectors Matri of a Linear Transformation Consider a linear ( transformation ) L : a b R 2 R 2 Suppose we know that L and L Then c d because of linearit, we can determine what L does
More informationChapter 3 Vectors. 3.1 Vector Analysis
Chapter 3 Vectors 3.1 Vector nalysis... 1 3.1.1 Introduction to Vectors... 1 3.1.2 Properties of Vectors... 1 3.2 Coordinate Systems... 6 3.2.1 Cartesian Coordinate System... 6 3.2.2 Cylindrical Coordinate
More informationUnit 12 Study Notes 1 Systems of Equations
You should learn to: Unit Stud Notes Sstems of Equations. Solve sstems of equations b substitution.. Solve sstems of equations b graphing (calculator). 3. Solve sstems of equations b elimination. 4. Solve
More informationMATRIX TRANSFORMATIONS
CHAPTER 5. MATRIX TRANSFORMATIONS INSTITIÚID TEICNEOLAÍOCHTA CHEATHARLACH INSTITUTE OF TECHNOLOGY CARLOW MATRIX TRANSFORMATIONS Matri Transformations Definition Let A and B be sets. A function f : A B
More informationComplex Numbers and Quaternions for Calc III
Complex Numbers and Quaternions for Calc III Taylor Dupuy September, 009 Contents 1 Introduction 1 Two Ways of Looking at Complex Numbers 1 3 Geometry of Complex Numbers 4 Quaternions 5 4.1 Connection
More informationChapter 1 Coordinates, points and lines
Cambridge Universit Press 978--36-6000-7 Cambridge International AS and A Level Mathematics: Pure Mathematics Coursebook Hugh Neill, Douglas Quadling, Julian Gilbe Ecerpt Chapter Coordinates, points and
More informationLecture 27: More on Rotational Kinematics
Lecture 27: More on Rotational Kinematics Let s work out the kinematics of rotational motion if α is constant: dω α = 1 2 α dω αt = ω ω ω = αt + ω ( t ) dφ α + ω = dφ t 2 α + ωo = φ φo = 1 2 = t o 2 φ
More informationP 0 (x 0, y 0 ), P 1 (x 1, y 1 ), P 2 (x 2, y 2 ), P 3 (x 3, y 3 ) The goal is to determine a third degree polynomial of the form,
Bezier Curves While working for the Renault automobile compan in France, an engineer b the name of P. Bezier developed a sstem for designing car bodies based partl on some fairl straightforward mathematics.
More informationMAT 1275: Introduction to Mathematical Analysis. Graphs and Simplest Equations for Basic Trigonometric Functions. y=sin( x) Function
MAT 275: Introduction to Mathematical Analsis Dr. A. Rozenblum Graphs and Simplest Equations for Basic Trigonometric Functions We consider here three basic functions: sine, cosine and tangent. For them,
More informationTransformations. Chapter D Transformations Translation
Chapter 4 Transformations Transformations between arbitrary vector spaces, especially linear transformations, are usually studied in a linear algebra class. Here, we focus our attention to transformation
More informationsin(α + θ) = sin α cos θ + cos α sin θ cos(α + θ) = cos α cos θ sin α sin θ
Rotations in the 2D Plane Trigonometric addition formulas: sin(α + θ) = sin α cos θ + cos α sin θ cos(α + θ) = cos α cos θ sin α sin θ Rotate coordinates by angle θ: 1. Start with x = r cos α y = r sin
More informationIntroduction to Vector Spaces Linear Algebra, Spring 2011
Introduction to Vector Spaces Linear Algebra, Spring 2011 You probabl have heard the word vector before, perhaps in the contet of Calculus III or phsics. You probabl think of a vector like this: 5 3 or
More informationMathematics Trigonometry: Unit Circle
a place of mind F A C U L T Y O F E D U C A T I O N Department of Curriculum and Pedagog Mathematics Trigonometr: Unit Circle Science and Mathematics Education Research Group Supported b UBC Teaching and
More informationDealing with Rotating Coordinate Systems Physics 321. (Eq.1)
Dealing with Rotating Coordinate Systems Physics 321 The treatment of rotating coordinate frames can be very confusing because there are two different sets of aes, and one set of aes is not constant in
More informationx y plane is the plane in which the stresses act, yy xy xy Figure 3.5.1: non-zero stress components acting in the x y plane
3.5 Plane Stress This section is concerned with a special two-dimensional state of stress called plane stress. It is important for two reasons: () it arises in real components (particularl in thin components
More informationPure Further Mathematics 2. Revision Notes
Pure Further Mathematics Revision Notes October 016 FP OCT 016 SDB Further Pure 1 Inequalities... 3 Algebraic solutions... 3 Graphical solutions... 4 Series Method of Differences... 5 3 Comple Numbers...
More informationLinear Equations and Arithmetic Sequences
CONDENSED LESSON.1 Linear Equations and Arithmetic Sequences In this lesson, ou Write eplicit formulas for arithmetic sequences Write linear equations in intercept form You learned about recursive formulas
More informationMathematics 309 Conic sections and their applicationsn. Chapter 2. Quadric figures. ai,j x i x j + b i x i + c =0. 1. Coordinate changes
Mathematics 309 Conic sections and their applicationsn Chapter 2. Quadric figures In this chapter want to outline quickl how to decide what figure associated in 2D and 3D to quadratic equations look like.
More informationChapter 3 Motion in a Plane
Chapter 3 Motion in a Plane Introduce ectors and scalars. Vectors hae direction as well as magnitude. The are represented b arrows. The arrow points in the direction of the ector and its length is related
More informationRising HONORS Algebra 2 TRIG student Summer Packet for 2016 (school year )
Rising HONORS Algebra TRIG student Summer Packet for 016 (school ear 016-17) Welcome to Algebra TRIG! To be successful in Algebra Trig, ou must be proficient at solving and simplifing each tpe of problem
More informationAPPENDIX D Rotation and the General Second-Degree Equation
APPENDIX D Rotation and the General Second-Degree Equation Rotation of Aes Invariants Under Rotation After rotation of the - and -aes counterclockwise through an angle, the rotated aes are denoted as the
More informationLinear Algebra March 16, 2019
Linear Algebra March 16, 2019 2 Contents 0.1 Notation................................ 4 1 Systems of linear equations, and matrices 5 1.1 Systems of linear equations..................... 5 1.2 Augmented
More informationHow to (Almost) Square a Circle
How to (Almost) Square a Circle Moti Ben-Ari Department of Science Teaching Weizmann Institute of Science http://www.weizmann.ac.il/sci-tea/benari/ c 2017 by Moti Ben-Ari. This work is licensed under the
More informationMathematics for Graphics and Vision
Mathematics for Graphics and Vision Steven Mills March 3, 06 Contents Introduction 5 Scalars 6. Visualising Scalars........................ 6. Operations on Scalars...................... 6.3 A Note on
More informationThe American School of Marrakesh. Algebra 2 Algebra 2 Summer Preparation Packet
The American School of Marrakesh Algebra Algebra Summer Preparation Packet Summer 016 Algebra Summer Preparation Packet This summer packet contains eciting math problems designed to ensure our readiness
More informationChapter 4 Analytic Trigonometry
Analtic Trigonometr Chapter Analtic Trigonometr Inverse Trigonometric Functions The trigonometric functions act as an operator on the variable (angle, resulting in an output value Suppose this process
More informationAnswer Explanations. The SAT Subject Tests. Mathematics Level 1 & 2 TO PRACTICE QUESTIONS FROM THE SAT SUBJECT TESTS STUDENT GUIDE
The SAT Subject Tests Answer Eplanations TO PRACTICE QUESTIONS FROM THE SAT SUBJECT TESTS STUDENT GUIDE Mathematics Level & Visit sat.org/stpractice to get more practice and stud tips for the Subject Test
More informationIntroduction to vectors
Lecture 4 Introduction to vectors Course website: http://facult.uml.edu/andri_danlov/teaching/phsicsi Lecture Capture: http://echo360.uml.edu/danlov2013/phsics1fall.html 95.141, Fall 2013, Lecture 3 Outline
More informationc) Words: The cost of a taxicab is $2.00 for the first 1/4 of a mile and $1.00 for each additional 1/8 of a mile.
Functions Definition: A function f, defined from a set A to a set B, is a rule that associates with each element of the set A one, and onl one, element of the set B. Eamples: a) Graphs: b) Tables: 0 50
More informationH.Algebra 2 Summer Review Packet
H.Algebra Summer Review Packet 1 Correlation of Algebra Summer Packet with Algebra 1 Objectives A. Simplifing Polnomial Epressions Objectives: The student will be able to: Use the commutative, associative,
More informationSection 3.1. ; X = (0, 1]. (i) f : R R R, f (x, y) = x y
Paul J. Bruillard MATH 0.970 Problem Set 6 An Introduction to Abstract Mathematics R. Bond and W. Keane Section 3.1: 3b,c,e,i, 4bd, 6, 9, 15, 16, 18c,e, 19a, 0, 1b Section 3.: 1f,i, e, 6, 1e,f,h, 13e,
More informationMathematics. Mathematics 2. hsn.uk.net. Higher HSN22000
Higher Mathematics UNIT Mathematics HSN000 This document was produced speciall for the HSN.uk.net website, and we require that an copies or derivative works attribute the work to Higher Still Notes. For
More informationVECTORS IN THREE DIMENSIONS
1 CHAPTER 2. BASIC TRIGONOMETRY 1 INSTITIÚID TEICNEOLAÍOCHTA CHEATHARLACH INSTITUTE OF TECHNOLOGY CARLOW VECTORS IN THREE DIMENSIONS 1 Vectors in Two Dimensions A vector is an object which has magnitude
More informationKinematics. Félix Monasterio-Huelin, Álvaro Gutiérrez & Blanca Larraga. September 5, Contents 1. List of Figures 1.
Kinematics Féli Monasterio-Huelin, Álvaro Gutiérre & Blanca Larraga September 5, 2018 Contents Contents 1 List of Figures 1 List of Tables 2 Acronm list 3 1 Degrees of freedom and kinematic chains of rigid
More informationProperties of Limits
33460_003qd //04 :3 PM Page 59 SECTION 3 Evaluating Limits Analticall 59 Section 3 Evaluating Limits Analticall Evaluate a it using properties of its Develop and use a strateg for finding its Evaluate
More informationMechanics Physics 151
Mechanics Phsics 151 Lecture 8 Rigid Bod Motion (Chapter 4) What We Did Last Time! Discussed scattering problem! Foundation for all experimental phsics! Defined and calculated cross sections! Differential
More informationInternational Examinations. Advanced Level Mathematics Pure Mathematics 1 Hugh Neill and Douglas Quadling
International Eaminations Advanced Level Mathematics Pure Mathematics Hugh Neill and Douglas Quadling PUBLISHED BY THE PRESS SYNDICATE OF THE UNIVERSITY OF CAMBRIDGE The Pitt Building, Trumpington Street,
More informationThe first change comes in how we associate operators with classical observables. In one dimension, we had. p p ˆ
VI. Angular momentum Up to this point, we have been dealing primaril with one dimensional sstems. In practice, of course, most of the sstems we deal with live in three dimensions and 1D quantum mechanics
More informationThree-Dimensional Coordinate Systems. Three-Dimensional Coordinate Systems. Three-Dimensional Coordinate Systems. Three-Dimensional Coordinate Systems
To locate a point in a plane, two numbers are necessary. We know that any point in the plane can be represented as an ordered pair (a, b) of real numbers, where a is the x-coordinate and b is the y-coordinate.
More informationCS 354R: Computer Game Technology
CS 354R: Computer Game Technolog Transformations Fall 207 Universit of Teas at Austin CS 354R Game Technolog S. Abraham Transformations What are the? Wh should we care? Universit of Teas at Austin CS 354R
More informationVectors Part 2: Three Dimensions
Vectors Part 2: Three Dimensions Last modified: 23/02/2018 Links Vectors Recap Three Dimensions: Cartesian Form Three Dimensions: Standard Unit Vectors Three Dimensions: Polar Form Basic Vector Maths Three
More informationINTRODUCTION GOOD LUCK!
INTRODUCTION The Summer Skills Assignment for has been developed to provide all learners of our St. Mar s Count Public Schools communit an opportunit to shore up their prerequisite mathematical skills
More information11.4 Polar Coordinates
11. Polar Coordinates 917 11. Polar Coordinates In Section 1.1, we introduced the Cartesian coordinates of a point in the plane as a means of assigning ordered pairs of numbers to points in the plane.
More informationCH. 1 FUNDAMENTAL PRINCIPLES OF MECHANICS
446.201 (Solid echanics) Professor Youn, eng Dong CH. 1 FUNDENTL PRINCIPLES OF ECHNICS Ch. 1 Fundamental Principles of echanics 1 / 14 446.201 (Solid echanics) Professor Youn, eng Dong 1.2 Generalied Procedure
More informationMatrices and Vectors
Matrices and Vectors James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University November 11, 2013 Outline 1 Matrices and Vectors 2 Vector Details 3 Matrix
More informationTable of Contents. Module 1
Table of Contents Module 1 11 Order of Operations 16 Signed Numbers 1 Factorization of Integers 17 Further Signed Numbers 13 Fractions 18 Power Laws 14 Fractions and Decimals 19 Introduction to Algebra
More informationSome linear transformations on R 2 Math 130 Linear Algebra D Joyce, Fall 2013
Some linear transformations on R 2 Math 3 Linear Algebra D Joce, Fall 23 Let s look at some some linear transformations on the plane R 2. We ll look at several kinds of operators on R 2 including reflections,
More informationVector (cross) product *
OpenStax-CNX module: m13603 1 Vector (cross) product * Sunil Kumar Singh This work is produced by OpenStax-CNX and licensed under the Creative Commons Attribution License 2.0 Abstract Vector multiplication
More informationUniversity of Alabama Department of Physics and Astronomy. PH 125 / LeClair Spring A Short Math Guide. Cartesian (x, y) Polar (r, θ)
University of Alabama Department of Physics and Astronomy PH 125 / LeClair Spring 2009 A Short Math Guide 1 Definition of coordinates Relationship between 2D cartesian (, y) and polar (r, θ) coordinates.
More information1 HOMOGENEOUS TRANSFORMATIONS
HOMOGENEOUS TRANSFORMATIONS Purpose: The purpose of this chapter is to introduce ou to the Homogeneous Transformation. This simple 4 4 transformation is used in the geometr engines of CAD sstems and in
More informationMathematics. Mathematics 2. hsn.uk.net. Higher HSN22000
hsn.uk.net Higher Mathematics UNIT Mathematics HSN000 This document was produced speciall for the HSN.uk.net website, and we require that an copies or derivative works attribute the work to Higher Still
More informationSYLLABUS FOR ENTRANCE EXAMINATION NANYANG TECHNOLOGICAL UNIVERSITY FOR INTERNATIONAL STUDENTS A-LEVEL MATHEMATICS
SYLLABUS FOR ENTRANCE EXAMINATION NANYANG TECHNOLOGICAL UNIVERSITY FOR INTERNATIONAL STUDENTS A-LEVEL MATHEMATICS STRUCTURE OF EXAMINATION PAPER. There will be one -hour paper consisting of 4 questions..
More information8.1 Exponents and Roots
Section 8. Eponents and Roots 75 8. Eponents and Roots Before defining the net famil of functions, the eponential functions, we will need to discuss eponent notation in detail. As we shall see, eponents
More informationA matrix over a field F is a rectangular array of elements from F. The symbol
Chapter MATRICES Matrix arithmetic A matrix over a field F is a rectangular array of elements from F The symbol M m n (F ) denotes the collection of all m n matrices over F Matrices will usually be denoted
More informationCS 335 Graphics and Multimedia. 2D Graphics Primitives and Transformation
C 335 Graphics and Multimedia D Graphics Primitives and Transformation Basic Mathematical Concepts Review Coordinate Reference Frames D Cartesian Reference Frames (a) (b) creen Cartesian reference sstems
More informationReview of Coordinate Systems
Vector in 2 R and 3 R Review of Coordinate Systems Used to describe the position of a point in space Common coordinate systems are: Cartesian Polar Cartesian Coordinate System Also called rectangular coordinate
More informationReview of Essential Skills and Knowledge
Review of Essential Skills and Knowledge R Eponent Laws...50 R Epanding and Simplifing Polnomial Epressions...5 R 3 Factoring Polnomial Epressions...5 R Working with Rational Epressions...55 R 5 Slope
More informationQUALITATIVE ANALYSIS OF DIFFERENTIAL EQUATIONS
arxiv:1803.0591v1 [math.gm] QUALITATIVE ANALYSIS OF DIFFERENTIAL EQUATIONS Aleander Panfilov stable spiral det A 6 3 5 4 non stable spiral D=0 stable node center non stable node saddle 1 tr A QUALITATIVE
More informationCSC Computer Graphics
CSC. Computer Graphics Lecture 4 P T P t, t t P,, P T t P P T Kasun@dscs.sjp.ac.lk Department o Computer Science Universit o Sri Jaewardanepura Kasun@dscs.sjp.ac.lk - Facult o Applied Sciences o USJP r
More informationCourse 2BA1: Hilary Term 2007 Section 8: Quaternions and Rotations
Course BA1: Hilary Term 007 Section 8: Quaternions and Rotations David R. Wilkins Copyright c David R. Wilkins 005 Contents 8 Quaternions and Rotations 1 8.1 Quaternions............................ 1 8.
More informationChapter 3 Vectors 3-1
Chapter 3 Vectors Chapter 3 Vectors... 2 3.1 Vector Analysis... 2 3.1.1 Introduction to Vectors... 2 3.1.2 Properties of Vectors... 2 3.2 Cartesian Coordinate System... 6 3.2.1 Cartesian Coordinates...
More informationUNCORRECTED SAMPLE PAGES. 3Quadratics. Chapter 3. Objectives
Chapter 3 3Quadratics Objectives To recognise and sketch the graphs of quadratic polnomials. To find the ke features of the graph of a quadratic polnomial: ais intercepts, turning point and ais of smmetr.
More informationAP Calculus AB Summer Assignment
AP Calculus AB Summer Assignment Name: When you come back to school, it is my epectation that you will have this packet completed. You will be way behind at the beginning of the year if you haven t attempted
More information74 Maths Quest 10 for Victoria
Linear graphs Maria is working in the kitchen making some high energ biscuits using peanuts and chocolate chips. She wants to make less than g of biscuits but wants the biscuits to contain at least 8 g
More informationTrigonometric Functions
Trigonometric Functions This section reviews radian measure and the basic trigonometric functions. C ' θ r s ' ngles ngles are measured in degrees or radians. The number of radians in the central angle
More information( ) ( ) ( ) ( ) TNM046: Datorgrafik. Transformations. Linear Algebra. Linear Algebra. Sasan Gooran VT Transposition. Scalar (dot) product:
TNM046: Datorgrafik Transformations Sasan Gooran VT 04 Linear Algebra ( ) ( ) =,, 3 =,, 3 Transposition t = 3 t = 3 Scalar (dot) product: Length (Norm): = t = + + 3 3 = = + + 3 Normaliation: ˆ = Linear
More informationLINEAR ALGEBRA - CHAPTER 1: VECTORS
LINEAR ALGEBRA - CHAPTER 1: VECTORS A game to introduce Linear Algebra In measurement, there are many quantities whose description entirely rely on magnitude, i.e., length, area, volume, mass and temperature.
More information