geometry in matlab slide

Geometry (Many slides adapted from Octavia Camps and Amitabh Varshney) Goals • Represent points, lines and triangles as column vectors. • Represent motion as matrices. • Move geometric objects with matrix multiplication. • Refresh memory about geometry and linear algebra Vectors • Ordered set of numbers: (1,2,3,4) • Example: (x,y,z) coordinates of pt in space. runit vecto a is ,1 If ),( 1 2 ,, 21 vv xv xxxv n i i n = = = ∑ =  Vector Addition ),(),(),( 22112121 yxyxyyxx ++=+=+ wv v v w w V+w V+w Scalar Product ),(),( 2121 axaxxxaa == v v v av av Inner (dot) Product v v w w α α 22112121 .),).(,(. yxyxyyxxwv +== The inner product is a The inner product is a SCALAR! SCALAR! α cos||||||||),).(,(. 2121 wvyyxxwv ⋅== wvwv ⊥⇔= 0. Points Using these facts, we can represent points. Note: (x,y,z) = x(1,0,0) + y(0,1,0) + z(0,0,1) x = (x,y,z).(1,0,0) y = (x,y,z).(0,1,0) z = (x,y,z).(0,0,1) Lines • Line: y = mx + a • Line: sum of a point and a vector P = P 1 + αd (whered is a column vector) • Line: Affine sum of two points P = α 1 P 1 + α 2 P 2 , where α 1 + α 2 = 1 Line Segment: For 0 ≤ α 1 , α 2 ≤ 1, P lies between P 1 and P 2 • Line: set of points equidistant from the origin in the direction of a unit vector. P 1 d P 1 P 2 Plane and Triangle • Plane: sum of a point and two vectors P = P 1 + αu + β v Plane: set of points equidistant from origin in direction of a vector. • Triangle: Affine sum of three points with α i ≥ 0 P = α 1 P 1 + α 2 P 2 + α 3 P 3 , where α 1 + α 2 + α 3 = 1 P lies between P 1 , P 2 , P 3 P 1 u v P 1 P 2 P 3 Generalizing Affine Sum of arbitrary number of points: Convex Hull P = α 1 P 1 + α 2 P 2 + … + α n P n , where α 1 + α 2 + … + α n = 1 and α i ≥ 0 [...]... c d     sin θ = − cosθ  sin θ = sy  − cosθ SVD cosθ   s x  0 sin θ    sx cosθ    sy sin θ  0  0  sin ϕ  s y − cos ϕ  0  sin ϕ ⋅  − cos ϕ 1  cos ϕ   x   ⋅  y sin ϕ    cos ϕ   x  sin ϕ   y    Affine Transformation  x a b tx    P' →  ⋅  y   c d ty   1    Viewing Position • Express world in new coordinate system • If origins same, this... coefficients of x, y, z in a plane equation define the normal Normal of a Triangle Normal of the plane containing the triangle (P1, P2 , P3 ): n P3 n = (P2 - P1) × (P3 - P1) P3 P2 Normal pointing towards you P1 P1 P2 Normal pointing away from you P2 P1 P3 • Models constructed with consistent ordering of triangle vertices : − all clockwise or all counter-clockwise • Usually normals point out of the model... satisfy the following constraints: R ⋅ RT = RT ⋅ R = I det(R ) = 1 Transformations can be composed • Matrix multiplication is associative • Combine series of transformations into one matrix (example, whiteboard) • In general, the order matters (example, whiteboard) • 2D Rotations can be interchanged Why? Rotation and Translation ( cos θ -sin θ tx sin θ cos θ ty 0 0 )( ) x y 1 1 Rotation, Scaling and Translation...  P Scaling P’ P Scaling Equation P’ s.y P P = ( x, y ) P ' = ( sx, sy ) P' = s ⋅ P y x s.x  sx   s 0  x  P' →   =  ⋅  y   sy  0 s    P' = S ⋅ P S Rotation P P’ Rotation Equations Counter-clockwise rotation by an angle θ Y’ P’ θ P y X’ x  x' cos θ  y ' =  sin θ    P' = R.P − sin θ   x  cosθ   y    Degrees of Freedom  x' cosθ  y ' =  sin θ    − sin θ  ... Rotation about an arbitrary point • Can translate to origin, rotate, translate back (example, whiteboard) • This is also rotation with one translation – Intuitively, amount of rotation is same either way – But a translation is added Stretching Equation P = ( x, y ) P ' = ( s x x, s y y ) P’ Sy.y  sx x  sx P' →   =  sy y  0  P y x 0   x ⋅   sy   y S Sx.x P' = S ⋅ P Linear Transformation a... counter-clockwise • Usually normals point out of the model Normal of a Vertex in a Mesh nv = (n1 +n2 + … +nk) / k = ∑ni / k = average of adjacent triangle normals ∀nv ∀nk or better: nv = ∑ (Αi ni) / (k ∑ (Αi) ) = area-weighted average of adjacent triangle normals ∀n ∀n1 3 ∀n2 Geometry Continued • Much of last classes’ material in Appendix A Matrices An×m  a11 a  21 = a31    an1  a12 a22...  y    Affine Transformation  x a b tx    P' →  ⋅  y   c d ty   1    Viewing Position • Express world in new coordinate system • If origins same, this is done by taking inner product with new coordinates • Otherwise, we must translate ... Matrices Determinant: A must be square  a11 det  a21 a12  a11 = a a22  21  a11 det a21  a31  a12 a22 a32 a12 = a11a22 − a21a12 a22 a13   = a a22 a23  11 a32 a33   a23 a21 − a12 a33 a31 a23 a21 + a13 a33 a31 a22 a32 Euclidean transformations 2D Translation P’ t P 2D Translation Equation ty P P’ t y x P = ( x, y ) t = (t x , t y ) tx P ' = ( x + t x , y + t y ) = P+t 2D Translation using Matrices...Normal of a Plane Plane: sum of a point and two vectors P = P1 + αu + βv v P - P1 = αu + βv P1 Ifn is orthogonal tou andv (n =u T T u ×v ) : T n • (P - P1) = αn •u + βn •v = 0 n v P1 u Implicit Equation of a Plane T n . P 3 P 1 P 2 P 3 P 1 P 3 P 2 Normal pointing towards you Normal pointing away from you • Models constructed with consistent ordering of triangle vertices : − all clockwise or all counter-clockwise. • Usually normals point. + a • Line: sum of a point and a vector P = P 1 + αd (whered is a column vector) • Line: Affine sum of two points P = α 1 P 1 + α 2 P 2 , where α 1 + α 2 = 1 Line Segment: For 0 ≤ α 1 ,. between P 1 and P 2 • Line: set of points equidistant from the origin in the direction of a unit vector. P 1 d P 1 P 2 Plane and Triangle • Plane: sum of a point and two vectors P = P 1