Lecture 02: 2D Geometry. August 24, 2017

Transcription

1 Lecture 02: 2D Geometry August 24, 207

2 This Came Up Tuesday Know what radiosity computation does. Do not expect to implement nor see underlying equations this semester. 8/24/7 CSU CS40 Fall 207, Ross Beveridge & Bruce Draper 2

3 Example Courtesy of Nikolay Radaev Rendered using 3D Studio Max (+VRay Plugin) 8/24/7 CSU CS40 Fall 207, Ross Beveridge & Bruce Draper 3

4 Now The Journey to 2D Rotation 8/24/7 CSU CS40 Fall 207, Ross Beveridge & Bruce Draper 4

5 Vectors, Points & Matrices The geometry for graphics rests upon Scalars, Vectors, Points, and Matrices And why? The short answer. Objects are collections of points Light rays are vectors Objects & Light interact in Euclidean spaces Placement in space is done using matrices Now for the longer answer 8/24/7 CSU CS40 Fall 207, Ross Beveridge & Bruce Draper 5

6 But let s start with Scalars Scalar - a number. Two Operations - Addition, Multiplication. Axioms Associative Commutative Invertible Invertible implies Subtraction Division α + β = β + α α β = β α α + ( β + γ ) = ( α + β) + γ α ( β γ ) = ( α β) γ α ( β + γ ) = α β + α γ 8/24/7 CSU CS40 Fall 207, Ross Beveridge & Bruce Draper 6

7 Vectors Vector - direction and magnitude Two Operations - Scalar-vector multiplication Vector-vector addition Often expressed as an n-tuple of scalars. v = v, v 2, v 3,,v n u v 2v v+u v 8/24/7 CSU CS40 Fall 207, Ross Beveridge & Bruce Draper 7

8 Test: Do you Get It? Are these two vectors the same? 8/24/7 CSU CS40 Fall 207, Ross Beveridge & Bruce Draper 8

9 Vector Spaces Linear combinations of vectors generate new vectors. u = α v + α 2 v 2 + α 3 v 3 for example u = 4 = or u = /24/7 CSU CS40 Fall 207, Ross Beveridge & Bruce Draper 9

10 Key Vector Space Concepts Span The space of all vectors that can be created by linear combinations of a set of vectors Basis Vectors A set of vectors that span a space Generally focus on basis vectors that are Orthogonal to each other (independent axes) Unit length Orthonormal Fancy term for a set of vectors that are orthogonal and unit length 8/24/7 CSU CS40 Fall 207, Ross Beveridge & Bruce Draper 0

11 Orthogonal? A B C D 8/24/7 CSU CS40 Fall 207, Ross Beveridge & Bruce Draper

12 Vectors beg Where are we? Directly over the center of the Earth? More seriously, vector spaces lack location Location requires an origin: a reference. Vector spaces have no origin. Affine spaces introduce an origin. They do this by introducing Points. New operations Point-point subtraction yields a vector. A point plus a vector yields a point. 8/24/7 CSU CS40 Fall 207, Ross Beveridge & Bruce Draper 2

13 Point + Vector = Point Linear combinations of basis vectors and a specified origin - a point. P = O + α v + α 2 v 2 + α 3 v 3 for example P = 4 = Typically, we think of the origin as being at [0,0,0], but that somewhat confuses the real meaning of an origin With an origin, you always know where you are (relatively). 8/24/7 CSU CS40 Fall 207, Ross Beveridge & Bruce Draper 3

14 But Wait, Do Points Exist Without Coordinates? The answer is yes! We are adopting the physicists view. Why does this matter In graphics, keeping the intrinsic geometry of objects separate from their coordinate manifestation in a particular frame of reference is essential. 8/24/7 CSU CS40 Fall 207, Ross Beveridge & Bruce Draper 4

15 Same Point - By Example Reference Frame A Reference Frame B The Sphere has intrinsic properties. Independent of reference frame A (or B). 8/24/7 CSU CS40 Fall 207, Ross Beveridge & Bruce Draper 5

16 Intrinsic Properties What matters is the relation of the data to the reference frame. Moving the sphere toward the reference point is the same as moving the reference point toward the sphere The same sphere can be expressed in different reference frames World Coordinates aren t special As long as all the data is expressed relative to the same reference frame 8/24/7 CSU CS40 Fall 207, Ross Beveridge & Bruce Draper 6

17 Where is a Point Revisited To specify a point in a Euclidean space. v 2 y Second Basis P P = O + x v + y v 2 An Origin O First Basis x v p x p y = x o y o + x 0 + y 0 8/24/7 CSU CS40 Fall 207, Ross Beveridge & Bruce Draper 7

18 A Point named Fred Fred Which is it? Fred = 6 6 or Fred = 4 3 8/24/7 CSU CS40 Fall 207, Ross Beveridge & Bruce Draper 8

19 2D Translation Think about the previous example Can you decide between Fred was moved down and to the left. Reference frame was moved up and to the right. Generally you cannot More important Often in graphics it is equally valid, or even preferable, to think of movement as shifting a reference frame rather than moving an object. 8/24/7 CSU CS40 Fall 207, Ross Beveridge & Bruce Draper 9

20 2D Translation - Moving Fred y v Fred u x x y = 6 6 and u v = 4 3 as translation 6 6 = /24/7 CSU CS40 Fall 207, Ross Beveridge & Bruce Draper 20

21 Affine to Euclidean Space Euclidean Space adds to affine space a new operation, the dot product. You all know the algebraic definition. u v = u i v i v = v v i Do you know its geometric interpretation? u v = u v cos( θ) If the dot product is zero, the vectors are orthogonal 8/24/7 CSU CS40 Fall 207, Ross Beveridge & Bruce Draper 2

22 Dot Product Yields Projection To start, set origin at zero. Now observe. p x p y = x 0 + y 0 x = p x p y 0, y = p x p y 0 The distance of a point from the origin along a dimension, i.e. along a basis vector, is measured by a dot product between the point and the basis vector 8/24/7 CSU CS40 Fall 207, Ross Beveridge & Bruce Draper 22

23 Know & Love Dot Products An easy way to define a line L F( x, y) = 0 n L ρ = 0 n n = v 2 P ρ = n P = n x p x + n y p y O r n n x n y x y ρ = 0 v n x x x + n y y ρ = 0 8/24/7 CSU CS40 Fall 207, Ross Beveridge & Bruce Draper 23

24 And in 3D Riddle: What do you call all points a distance of 3 from the origin measured in a direction defined by a vector n? 8/24/7 CSU CS40 Fall 207, Ross Beveridge & Bruce Draper 24

25 Further Dot Product Motivation Above you see how almost all texts and courses introduction 2D rotation. This is entirely correct, but there is a more intuitive way to understand rotation. 8/24/7 CSU CS40 Fall 207, Ross Beveridge & Bruce Draper 25

26 Know & Love Dot Products 2 Consider an alternate basis y P p x p y = x o y o + u v 2 2 v 2 O x v u v = x y + x o y o 8/24/7 CSU CS40 Fall 207, Ross Beveridge & Bruce Draper 26

27 Welcome to 2D Rotation x o y o = 0 0 Put origin at zero zero (common) u v = x y These are the same! u v = cos(45o ) sin(45 o ) sin(45 o ) cos(45 o ) x y 8/24/7 CSU CS40 Fall 207, Ross Beveridge & Bruce Draper 27

28 Rotate by q M = RP " R = $ $ # cos θ sin θ ( ) sin( θ) ( ) cos( θ) % ' ' &! P = # "# x y $ & %& écos( q) x - sin( q) yù ë ê sin( q) x + cos( q) yû ú = écos( q) - sin( q) ù ë ê sin( q ) cos( q) û ú é ë ê x y ù û ú q Does this make sense, given the geometry of the dot product? 8/24/7 CSU CS40 Fall 207, Ross Beveridge & Bruce Draper 28

29 More Standard Approach Derivation of Rotation Matrix x = r cos( θ) y = rsin( θ) x 2 = r cos( θ +φ) y 2 = rsin( θ +φ)! # "# x 2 y 2 $ & %& j q! # "# x y $ & %& 8/24/7 CSU CS40 Fall 207, Ross Beveridge & Bruce Draper 29

30 Derivation (cont.) cos a + b Magic Trig Identity: ( ) = cos a ( ) = sin a sin a + b x 2 = r cos θ +φ ( ) ( )cos b ( )cos b ( ) sin a ( ) + sin b ( )sin b ( )cos a ( ) ( ) x 2 = r cos θ ( )cos φ ( ) rsin θ ( )sin φ ( ) x 2 = x cos φ ( ) y sin φ ( ) Note: the process for y 2 is the same 8/24/7 CSU CS40 Fall 207, Ross Beveridge & Bruce Draper 30

31 The End