2D Geometry - points and polygons A point in 2D is represented by two real numbers (X,Y) Y X A line segment is represented by its two end points (X1,Y1) (X2,Y2), or a 2x2 matrix [ ] X1 Y1 X2 Y2 Y X

[ ] A polygon is represented by an list of points (X1,Y1), (X2,Y2), ..... (Xn,Yn) or a n x 2 matrix [ ] For example, a triangle is represent by [ ] X1 Y1 : Xn Yn X1 Y1 X2 Y2 X3 Y3 Y X

Object transformation and Coordinates System transformation Object transformation is different from the coordinates system transformation 2 1 8 Y X (2,4) (6,2) (2,2)

Object Transformation Y Scale (1/2) about the origin X Y Translation (2,1) X Rotation (p/2) counter clock wise about the origin Y X

Coordinates system transformation Scale (1/2) about the origin X* 1 2 3 Y Translation (2,1) X Y Rotation (p/2) counter clock wise about the origin X If not described explicitly, transformation always means object transformation

2D Translations There are 3 "basic" transformations: (1) Translation (2) Scaling (3) Rotation A series of transformations can be combined (concatenated) into one. (1) Translation : T(Tx, Ty) Y (x',y') Ty (x,y) Tx X

(2) Scaling (about the origin) : S(Sx, Sy) What about "Mirror Images"? How do we avoid distortion? What happens when the scale factor equal to zero? X Y (x,y) (x',y') S(-2, -1)

(3) Rotation (about the origin counterclockwise) : x = R cos a y = R sin a x' = R cos(a + q) = R(cos a cos q - sin a sin q) y' = R sin(a + q) = R(sin a cos q + cos a sinq) So, x ' = xcosq - ysinq y ' = xsinq + ycosq Y (x',y') (x,y) q a X

Example: scaling about arbitrary point. Concatenation -- A series of transformations can be combined (concatenated) into one. Example: scaling about arbitrary point. 1. Translate so that point (a,b) becomes the temporary origin: x1 = x - a y1 = y - b 2. Scale to the correct size: x2 = Sx*x1 y2 = Sy*y1 3. Translate again to restore the coordinates of (a,b): x3 = x2 + a y3 = y2 + b Y Hence, we can concatenate any number of basic transformations into one set of equations, eliminating the intermediate results -- which would have required extra data storage and computation. (x,y) (a,b) X

Algebraic representations for transformations are very limited: (1) A special procedure is need for each basic transformation and other known concatenated forms. (2) It is difficult to provide general transformation capabilities in algebraic form in the computer. (3) How can one find the inverse of a concatenated equation (to restore the original position, for example)?

[ ] Let the point (x,y) be a row vector [x y]: x' = ax + by and y' = cx + dy can be expressed in the matrix equation: [x' y'] = [x y] * [ ]2x2 Let P' = [x' y'] and P = [x y], this becomes the matrix equation: P' = P * T where P' =[x' y'] , P = [x y] , T = Consider three basic transformations, can we find a "T" for each? What about translations? X'=X+Tx, Y'=Y+Ty No, P' =[x' y']=[x y] [ ]+[Tx Ty]=P*T+Q What about scaling? X'=Sx*X, Y'=Sy*Y Yes, P' =[x' y']=[x y] [ ] =P*T What about rotation? x'=xcosq-ysinq, y'=xsinq+ycosq Yes, P' =[x' y']=[x y] [ ] [ ] a c b d 1 0 0 1 Sx 0 0 Sy cosq sinq -sinq cosq

Homogeneous Coordinates Let the point (x,y) be a row vector [x y 1]: x' = ax + by + e and y' = cx + dy + f can be expressed in the matrix equation: [x' y' 1] = [x y 1] [ ]3x3 This becomes the matrix equation P' = P * T where P'=[x' y' 1], P= [x y 1], T= [ ] Now, can we find a "T" for each transformation? What about translations? Yes, T= [ ] What about scaling? Yes, T= [ ] What about rotation? Yes, T= [ ]

To check, what should the matrix be if we: (1) translate by (0,0) ? P'=P*[ ]=P*[ ]=P (2) scale by (1,1) ? (3) rotate by 0 o ? P'=P*[ ]=P*[ ]=P (4) rotate by 90o ? P'=[x' y' 1]=[x y 1] [ ] =[1 0 1]*[ ]=[0 1 1] Draw the pictures and find the new vertices of the triangle for the following transformations (See the figure below) : (x, y) = (1, 0) (x', y') = (0, 1) X Y (2,2) (2,1) (4,1) A B C

(5) translate A by (2,1). (6) translate A by (-2,1). (7) scale A by (2,1). (8) scale A by (-1,1). X Y (2,2) (2,1) (4,1) A B C X Y (2,2) (2,1) (4,1) A B C X Y (2,2) (2,1) (4,1) A B C Y A (2,2) C (4,1) B (2,1) X

(9) scale A by (1,0). (10) scale A by (0,0). (11) rotate A by 0o. X Y (2,2) (2,1) (4,1) A B C X Y (2,2) (2,1) (4,1) A B C X Y (2,2) (2,1) (4,1) A B C Y A (2,2) C (4,1) B (2,1) X

Concatenation is easy now. Consider our scaling example: Example: We want to reduce the size or the box, without changing the position of point (a, b) Functional form and matrix form for each transformations: (1) translate: T(-a,-b) (2) scale : S(Sx,Sy) (3) translate: T(a,b) Composite 2D Transformation (a,b) Y (x, y) X

Combining the matrix equations: P3 =P2* T (-a,-b)=(P1*S(Sx, Sy) )*T(a, b) =P*T(-a, -b) *S(Sx, Sy)*T(a, b) Multiply the matrices : Now, the matrix equation is : or, in algebraic form : Thus, a single 3x3 matrix can represent any combination of basic transformations in a simple form.