2D Geometry. Tuesday September 2 rd, 2014.

Transcription

1 D Geometry Tuesday September rd, 04.

2 Scalars, Vectors & Points! The geometry for graphics rests upon! Scalars, Vectors, Points! And why? One simple answer.! Objects and viewpoints are built of points.! Points depend upon vectors.! Vectors depend upon scalars.! Better reason takes more time! We are going that direction. 8/8/4 CS 40 Ross Beveridge & Bruce Draper

3 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 8/8/4 CS 40 Ross Beveridge & Bruce Draper Slide 3

4 Vectors! Vector - direction and magnitude! Two Operations -! Vector-vector addition! Scalar-vector multiplication! Often expressed as an n-tuple of scalars. v = v, v, v 3,,v n v v v+u u v 8/8/4 CS 40 Ross Beveridge & Bruce Draper Slide 4

5 A bit of this, a bit of that.! Linear combinations of vectors.! Generates new vectors. u = α v + α v + α 3 v 3 for example u = 4 = or u = Here is the essential concept of basis vectors. 8/8/4 CS 40 Ross Beveridge & Bruce Draper Slide 5

6 Vectors beg Where are we? Directly over the center of the Earth?! More seriously,! Location requires an origin: a reference.! Affine spaces introduce this origin.! They do this by introducing Points.! Point-point subtraction yields a vector.! A point plus a vector yields a point. 8/8/4 CS 40 Ross Beveridge & Bruce Draper Slide 6

7 Point + Vector = Point! Linear combinations of basis vectors! and a specified origin - a point. P = O + α v + α v + α 3 v 3 for example P = 4 = Typically we think of the origin as being at the point [0, 0, 0], but that somewhat confuses the real meaning of an origin. With an origin, you always know where you are. 8/8/4 CS 40 Ross Beveridge & Bruce Draper Slide 7

8 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. 8/8/4 CS 40 Ross Beveridge & Bruce Draper Slide 8

9 Same Point - By Example Reference Frame A Reference Frame B! The Sphere has intrinsic properties.! Independent of reference frame A (or B). 8/8/4 CS 40 Ross Beveridge & Bruce Draper Slide 9

10 Affine to Euclidean Space! Euclidean Space adds to affine space a new operation, the dot product.! You all know the algebraic definition. n u v = u i v i v = v v u v = u v cos θ i u v = 0 iff u and v are orthogonal You may not have thought about how central the dot product is to the notion of location, basis vectors and transformations. ( ) 8/8/4 CS 40 Ross Beveridge & Bruce Draper Slide 0

11 Where is a Point Revisited! To specify a point in a Euclidean space. v y Second Basis P P = O + x v + y v An Origin O First Basis x v p x p y = x o y o + x 0 + y 0 Recall point-vector addition. Spot the dot product yet? 8/8/4 CS 40 Ross Beveridge & Bruce Draper Slide

12 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. x = P v y = P v 8/8/4 CS 40 Ross Beveridge & Bruce Draper Slide

13 Know & Love Dot Products! An easy way to define a line L F( x, y) = 0 n L ρ = 0 n n = v P ρ = n P = n x p x + n y p y O ρ n n x n y x y ρ = 0 v n x x x + n y y ρ = 0 8/8/4 CS 40 Ross Beveridge & Bruce Draper Slide 3

14 An Aside: Works in 3D Also 8/8/4 CS 40 Ross Beveridge & Bruce Draper Slide 4

15 Know & Love Dot Products! Consider an alternate basis y P p x p y = x o y o + u + v v O x v u v = x y + x o y o We are close to something very familiar! 8/8/4 CS 40 Ross Beveridge & Bruce Draper Slide 5

16 Welcome to D Rotation x o y o = 0 0 Put origin at zero zero (common) These are the same! u v u v = x y = cos(45o ) sin(45 o ) sin(45 o ) cos(45 o ) x y 8/8/4 CS 40 Ross Beveridge & Bruce Draper 6

17 Rotate by θ Trig. Approach cos ( M = R P, R = θ ) - sin ( θ ), sin ( θ ) cos ( θ ) P = x y cos ( θ ) x - sin ( θ ) y sin ( θ ) x + cos ( θ ) y = cos ( θ ) - sin ( θ ) sin ( θ ) cos ( θ ) x y θ Does this make sense, given the geometry of multiplication? 8/8/4 CS 40 Ross Beveridge & Bruce Draper 7

18 Derivation of Rotation Matrix x = rcos( θ) y = rsin( θ) x = rcos( θ + φ) y = rsin( θ + φ) x y ϕ θ x y 8/8/4 CS 40 Ross Beveridge & Bruce Draper 8

19 Derivation (cont.) Magic Trig Identity: cos( a + b) = cos( a)cos b sin(a + b) = sin( a)cos b x = r cos( θ +ϕ) ( ) sin a ( ) + sin b ( )sin b ( )cos a ( ) ( ) x = r cos( θ)cos( ϕ) r sin( θ)sin( ϕ) x = x cos( ϕ) y sin( ϕ) Note: the process for y is the same 8/8/4 CS 40 Ross Beveridge & Bruce Draper 9

20 The End