Vectors Coordinate frames 2D implicit curves 2D parametric curves. Graphics 2008/2009, period 1. Lecture 2: vectors, curves, and surfaces
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1 Graphics 2008/2009, period 1 Lecture 2 Vectors, curves, and surfaces
2 Computer graphics example: Pixar (source:
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17 Computer graphics example: Pixar (source:
18 Computer graphics example: Pixar Modeling Lighting Animating (source: Shading
19 Recap Computer graphics: Modelling (creating 3d virtual worlds) Rendering (creating shaded 2d images from 3d models) Two basic methods: Projective methods Ray tracing Last time: Lines/rays and spheres Intersections between them Today: More mathematics to describe objects
20 : definition Vectors Length Addition Bases Dot product and cross product A vector is specified by a length and direction In R d it can be defined as v 1 v 2 an ordered d-tuple v =.. v d v ( ) 5 v = 1
21 : definition Vectors Length Addition Bases Dot product and cross product v 1 In R 3, for example: v = v 2 v 3 v x x v or v y, or y v, v z z v or (v 1, v 2, v 3 ), or (v 1, v 2, v 3 ) T, or... v ( ) 5 v = 1
22 Vectors: algebraic interpretation Length Addition Bases Dot product and cross product A 2D vector (x v, y v ) can be seen as the point (x v, y v ) in the Cartesian plane. v ( ) 5 v = 1
23 Vectors: geometric interpretation Length Addition Bases Dot product and cross product A 2D vector (x v, y v ) can be seen as an offset from the origin. Such an offset (arrow) can be translated. v v v v = ( ) 5 1
24 : length Vectors Length Addition Bases Dot product and cross product The Euclidean length of a d-dimensional vector v is v = v1 2 + v v2 d v ( ) 5 v = 1
25 Vectors: scalar multiple Length Addition Bases Dot product and cross product A scalar multiple of a d-dimensional vector v is λv = (λv 1, λv 2,..., λv d ) Note that v and λv have the same direction or opposite directions
26 Parallel vectors Vectors Length Addition Bases Dot product and cross product Two vectors v 1 and v 2 are parallel if one is a scalar multiple of the other, i.e., there is a λ such that v 2 = λv 1. Note that if one of the vectors is the null vector, then the vectors are considered neither parallel nor not parallel.
27 Unit vectors Vectors Length Addition Bases Dot product and cross product A vector v is a unit vector if v = 1. Normalization Q: given an arbitrary vector v, how do we find a unit vector parallel to v? Can every vector be normalized?
28 Addition of vectors Vectors Length Addition Bases Dot product and cross product Given two vectors in R d, v = (v 1, v 2,..., v d ) and w = (w 1, w 2,..., w d ) their sum is defined as v + w = (v 1 + w 1, v 2 + w 2,..., v d + w d )
29 Addition of vectors Vectors Length Addition Bases Dot product and cross product Addition of vectors is commutative, as can be seen easily from the geometric interpretation. Q: show algebraically that vector addition is commutative. Q: what is the relation between v, w, and v + w?
30 Addition of vectors Vectors Length Addition Bases Dot product and cross product Q: How would subtraction be defined?
31 Bases Vectors Length Addition Bases Dot product and cross product A 2D vector can be expressed as a combination of any pair of non-parallel vectors. For instance, in the image, a = 1.5v + 0.6w. Such a pair is called linearly independent, and they form a 2D basis. The extension to higher dimensions is straightforward. v w a
32 Orthonormal basis Vectors Length Addition Bases Dot product and cross product Two vectors form an orthonormal basis if (1) they are orthogonal to each other, and (2) they are unit vectors. The advantage of an orthonormal basis is that lengths of vectors, expressed in the basis, are easy to calculate.
33 Length Addition Bases Dot product and cross product Dot product (inner product, scalar product) For two vectors v, w R d, the dot product is defined as v w = v 1 w 1 +v 2 w 2 + +v d w d, or v w = d i=1 v iw i We have that cos θ = v w v w, where θ is the angle between the two vectors. dot product
34 Length Addition Bases Dot product and cross product Dot product (inner product, scalar product) dot product
35 Length Addition Bases Dot product and cross product Dot product (inner product, scalar product) Q: what is the inner product of an arbitrary unit vector with itself? Q: what do we know if for two vectors v and w we have that v w = 0? dot product
36 Length Addition Bases Dot product and cross product Dot product (inner product, scalar product) dot product
37 Length Addition Bases Dot product and cross product Dot product (inner product, scalar product) Another use: Calculate the length of the projection of one vector onto another. dot product
38 Cross product Vectors Length Addition Bases Dot product and cross product For two vectors v, w R 3, the cross product is defined as v 2 w 3 v 3 w 2 v w = v 3 w 1 v 1 w 3 v 1 w 2 v 2 w 1 cross product We have that v w = v w sin θ, where θ is the angle between v and w.
39 Cross product Vectors Length Addition Bases Dot product and cross product Q: show that v w is orthogonal to both v and w. cross product
40 Cross product Q: what is v v? Vectors Length Addition Bases Dot product and cross product Q: what is v w if v and w are parallel? cross product
41 Cross product Vectors Length Addition Bases Dot product and cross product Q: is it possible or necessary that v and w are orthogonal to form v w? cross product
42 Left- and right-handed systems Bases and coordinate frames Left- and right-handed systems Coordinate systems in 3D come in two flavors: left-handed and right-handed. There are arguments for both left- and right-handed systems for The global system The camera system Object systems Z X Y Y Y Z X X Z Camera coordinate system Z Y Y X X Z World coordinate system
43 Left- and right-handed systems Bases and coordinate frames Coordinate transformations A frequent operation in graphics is the change from one coordinate system (e.g., the (u, v, w) camera system) to another (e.g., the (x, y, z) global system). v u w Having orthonormal bases for both systems makes the transformations simpler. Z Y X
44 General curves Circles Lines 2D Implicit curves f(x, y) y An implicit curve in 2D has the form x f(x, y) = 0 f maps two-dimensional points to a real value; the points for which this value is 0 are on the curve, while other points are not on the curve. y f(x, y) = 0 x
45 General curves Circles Lines Implicit representation of circles The implicit representation of a 2D circle with center c and radius r is p (x x c ) 2 + (y y c ) 2 r 2 = 0 c So for any point p that lies on the circle, we have that (p c) (p c) r 2 = 0, so p c 2 r 2 = 0, which gives p c = r.
46 General curves Circles Lines Implicit representation of circles The implicit representation of a 2D circle with center c and radius r is (x x c ) 2 + (y y c ) 2 r 2 = 0 So for any point p that lies on the circle, we have that (p c) (p c) r 2 = 0, so p c 2 r 2 = 0, which gives p c = r.
47 General curves Circles Lines Implicit representation of lines A well-known representation of lines is the slope-intercept form y = ax + b This can easily be converted to ax + y b = 0. If b = 0, the line intersects the origin, and we have ( ) x n p = 0, with p = and y ( ) a n = 1
48 General curves Circles Lines Implicit representation of lines n p = 0, with p = ( ) x, n = y ( ) a 1 Q: what if the line does not intersect the origin?
49 General curves Circles Lines Implicit representation of lines ( ) a Note: The vector n = 1 is orthogonal to the line. (This is true for the first 2 coefficiants of any implicit line representation.)
50 General curves Circles Lines Implicit representation of lines
51 General curves Circles Lines parametric implicit representation A parametric curve is controlled by a single parameter, and has the form ( ) x = y ( ) g(t) h(t) Parametric representations have some advantages over functions, even if a function would suffice to represent the curve. h(t) g(t)
52 General curves Circles Lines parametric implicit representation Parametric equation of a circle The parametric equation of a 2D circle with center c and radius r is ( ) x y = ( ) xc + r cos φ y c + r sin φ p φ c
53 General curves Circles Lines parametric implicit representation Parametric equation of a line The parametric equation of a line through the points p 0 and p 1 is ( ) x = y ( ) xp0 + t(x p1 x p0 ) y p0 + t(y p1 y p0 ) As we have seen before, this can alternatively be written as p(t) = p 0 + t(p 1 p 0 ). p 0 p 1
54 General curves Circles Lines parametric implicit representation Conversion between representations It is convenient to be able to convert a parametric equation of a line into an implicit equation, and vice versa. Q: how do we do that? p 0 p 1
55 General curves Circles Lines parametric implicit representation Conversion between representations Q: how do we do that? Slope-intercept repr.: y = ax + b Implicit representation: f(x, y) = ax + by + c = 0. Parametric representation (vector form): p(t) = p 0 + t(p 1 p 0 ) (Note: These are not the same a,b s for the different equations!) p 0 p 1
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