AML710 CAD LECTURE 5 2D TRANSFORMATIONS (Contd.) Sequence of operations, Matrix multiplication, concatenation, combination of operations Types of Transformation Affine Map: A map φ that maps E3 into itself is called an affine Map if it leaves barycentric conditions invariant. If 3 x = ∑ β j a j, x,a j ∈ E Then, 3 φx = ∑ β jφa j φx,φa j ∈ E Most of the transformations that are used to position or scale an object in CAD are affine maps. The name was given by L. Euler and studied systematically by A. Mobius Euclidian Maps: Rigid body motions like rotation and translation where lengths and angles are unchanged are called Euclidian maps. This is a special case of affine maps 1 Transformation Groups and Symmetries Affine transformations are classified (Felix Klein) as follows Similarity Groups: Rigid motion and scaling Eg: Congruent, similar Symmetry Groups:Rotation, Reflection, Translation 1. No translation and no reflection – Circle group Eg: a gear wheel 2. No translation and at least one reflection – Dihedral group 3. Translation along one direction 4. Translation along more than one direction Homogeneous coordinates of vertices ¾ A point in homogeneous coordinates (x, y, h), h ≠ 0, corresponds to the 2-D vertex (x/h, y/h) in Cartesian coordinates. ¾ Conceive that the Cartesian coordinates axes lies on the plane of h = 1. The intersection of the plane and the line connecting the origin and (x, y, h) gives the corresponding Cartesianw coordinates. (x, y, h) y (x/h, y/h, 1) y x h = 1 x h = 0 2 ¾ For example, both the points (6, 9, 3) and (4, 6, 2) in the homogeneous coordinates corresponds to (2, 3) in the Cartesian coordinates. Conversely, the point (2, 1) of the Cartesian corresponds to (2, 1, 1), (4, 2, 2) or (6, 3, 3) of the homogeneous system h 6, 3, 3) (4, 2, 2) y y (2, 1, 1) x h = 1 x h = 0 Rotation Consider the following figure where a position vector p(x,y) which makes an angle φ to x-axis after transformation to p’(x’,y’) makes an angle φ+θ degrees. P’(x’,y’) P(x,y) P= [x y] = [rcosφ rsinφ] θ P’= [x’ y’] = [rcos(φ+θ ) rsin(φ+θ )] φ = [r(cosφcosθ -sinφsinθ) r(cosφsinθ+ sinφcosθ) ] Using the definition of x=rcosφ and y=rsinφ, we can write P’= [x’ y’] = [x cosθ –y sinθ x sinθ +y cosθ ] or the transformation matrix is ⎡ cosθ sinθ ⎤ [][][X T = x y ]⎢ ⎥ ⎣− sinθ cosθ ⎦ 3 Some Common Cases of Rotation Rotation of 90° counter clockwise about the origin ⎡ cosθ sinθ ⎤ ⎡ 0 1⎤ []T = ⎢ ⎥ = ⎢ ⎥ ⎣− sinθ cosθ ⎦ ⎣−1 0⎦ Rotation of 180° counter clockwise about the origin ⎡ cosθ sinθ ⎤ ⎡−1 0 ⎤ []T = ⎢ ⎥ = ⎢ ⎥ ⎣− sinθ cosθ ⎦ ⎣ 0 −1⎦ Rotation of 270° counter clockwise about the origin ⎡ cosθ sinθ ⎤ ⎡0 −1⎤ []T = ⎢ ⎥ = ⎢ ⎥ ⎣− sinθ cosθ ⎦ ⎣1 0 ⎦ In all the above cases det[T]=1 Reflection- Special case of Rotation Reflection is a special case of rotation of 180° about a line in xy plane passing through the origin. Eg about y=0 (x-axis) ⎡1 0 ⎤ []T = ⎢ ⎥ ⎣0 −1⎦ About x=0 (y-axis) ⎡−1 0⎤ []T = ⎢ ⎥ ⎣ 0 1⎦ About the lines x=y and x-y respectively are: ⎡0 1⎤ ⎡ 0 −1⎤ []T = ⎢ ⎥ []T = ⎢ ⎥ ⎣1 0⎦ ⎣−1 0 ⎦ 4 Reflection- Properties If two pure reflections about a line passing through the origin are applied successively the result is a pure rotation. The determinant of a pure reflection matrix is -1 ⎡−1 0⎤ []T = ⎢ ⎥ ⎣ 0 1⎦ Properties of transformation Matrices Det[T]=? In case of a) rotation, b) reflection. What is the geometrical interpretation of inverse of [T]? Show that TT-1 = [I] ⎡ cosθ sinθ ⎤ []T = ⎢ ⎥ ⎣− sinθ cosθ ⎦ −1 ⎡ cos(−θ ) sin(−θ )⎤ ⎡cosθ − sinθ ⎤ T []T = ⎢ ⎥ = ⎢ ⎥ = T ⎣− sin(−θ ) cos(−θ)⎦ ⎣sinθ cosθ ⎦ Thus for a pure rotation (det[T]=1) the inverse of T is its transpose 5 Example Show that the following transformation matrix gives a pure rotation ⎡1− t 2 2t ⎤ ⎢ 2 2 ⎥ 1+ t 1+ t [T ] = ⎢ 2 ⎥ ⎢ − 2t 1− t ⎥ ⎣⎢1+ t 2 1+ t 2 ⎦⎥ ⎛1− t 2 ⎞⎛1− t 2 ⎞ ⎛ − 2t ⎞⎛ 2t ⎞ det[T] = ⎜ ⎟⎜ ⎟ − =1 ⎜ 2 ⎟⎜ 2 ⎟ ⎜ 2 ⎟⎜ 2 ⎟ ⎝1+ t ⎠⎝1+ t ⎠ ⎝1+ t ⎠⎝1+ t ⎠ Example A A unit square is transformed by a 2×2 transformation matrix. The resulting position vectors are: ⎡0 0⎤ ⎢2 3⎥ [x′] = ⎢ ⎥ ⎢8 4⎥ ⎢ ⎥ ⎣6 1⎦ Determine the transformation matrix used. ⎡0 0⎤ ⎡ 0 0 ⎤ ⎡0 0⎤ Ans: ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢1 0⎥⎡a b⎤ ⎢ a b ⎥ ⎢2 3⎥ ⎡2 3⎤ [x′] = [X ][T ] = ⎢ ⎥ = = ;∴[T ] = ⎢ ⎥ ⎢1 1⎥⎣c d⎦ ⎢a + c b + d⎥ ⎢8 4⎥ ⎣6 1⎦ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣0 1⎦ ⎣ c d ⎦ ⎣6 1⎦ 6 Example Show that the shear transformation in x and y directions together is not the same as shear along x followed by shear along y? Ans: ⎡1 b⎤⎡1 0⎤ ⎡1+ bc b⎤ ⎡1 b⎤ ⎢ ⎥⎢ ⎥ = ⎢ ⎥ ≠ ⎢ ⎥ ⎣0 1⎦⎣c 1⎦ ⎣ c 1⎦ ⎣c 1⎦ ⎡1 0⎤⎡1 b⎤ ⎡1 b ⎤ ⎡1 b⎤ ⎢ ⎥⎢ ⎥ = ⎢ ⎥ ≠ ⎢ ⎥ ⎣c 1⎦⎣0 1⎦ ⎣c 1+ bc⎦ ⎣c 1⎦ Example Problem • Consider a triangle whose vertices are (2 2), (4 2) and (4 4). Find the concatenated transformation matrix and the transformed vertices for rotatation of 90 about the origin followed by reflection through the line y = -x. Comment on the sequence of transformations. ⎡2 2⎤ ⎡2 2⎤ ⎡− 2 2⎤ ⎡ 0 1⎤ ⎡ 0 −1⎤ ⎡−1 0⎤ [X}[T }[T } = [X}[T} = ⎢4 2⎥ = ⎢4 2⎥ = ⎢− 4 2⎥ 1 2 ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎣−1 0⎦ ⎣−1 0 ⎦ ⎣ 0 1⎦ ⎣⎢4 4⎦⎥ ⎣⎢4 4⎦⎥ ⎣⎢− 4 4⎦⎥ For seeing the effect of changing the sequence of operations let us reverse the order, i.e, first reflection and then rotation. ⎡2 2⎤ ⎡2 2⎤ ⎡2 − 2⎤ ⎡ 0 −1⎤⎡ 0 1⎤ ⎡1 0 ⎤ [X}[T }[T } = [X}[T} = ⎢4 2⎥ = ⎢4 2⎥ = ⎢4 − 2⎥ 2 1 ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎣−1 0 ⎦⎣−1 0⎦ ⎣0 −1⎦ ⎣⎢4 4⎦⎥ ⎣⎢4 4⎦⎥ ⎣⎢4 − 4⎦⎥ 7 Example Problem: Solution ¾ Conduct a combination of transformations in sequence (1,3) ¾ Translate the right-angle vertex to the origin (Tx = -1, Ty = -1) (5,1) (1,1) ⎡ x'⎤ ⎡1 0 Tx ⎤⎡ x⎤ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ y'⎥ = ⎢0 1 Ty ⎥⎢ y⎥. ⎣⎢w'⎦⎥ ⎣⎢0 0 1 ⎦⎥⎣⎢w⎦⎥ (0,2) o π (4,0) ¾ Rotate 45 ( /4 radian) (0,0) π π sin /4 = cos /4 = 0.7071 ⎡ x"⎤ ⎡cosθ − sinθ 0⎤⎡ x'⎤ ⎢ ⎥ ⎢ ⎥⎢ ⎥ (2.8,2.8) ⎢ y"⎥ = ⎢sinθ cosθ 0⎥⎢ y'⎥ (-1.4,1.4) ⎣⎢w"⎦⎥ ⎣⎢ 0 0 1⎦⎥⎣⎢w'⎦⎥ 8 ⎡ x"⎤ ⎡cosθ − sinθ 0⎤⎡ x'⎤ ⎡cosθ − sinθ 0⎤⎡1 0 Tx ⎤⎡ x⎤ ⎢ y"⎥ = ⎢sinθ cosθ 0⎥⎢ y'⎥ = ⎢sinθ cosθ 0⎥⎢0 1 T ⎥⎢ y⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ y ⎥⎢ ⎥ ⎣⎢w"⎦⎥ ⎣⎢ 0 0 1⎦⎥⎣⎢w'⎦⎥ ⎣⎢ 0 0 1⎦⎥⎣⎢0 0 1 ⎦⎥⎣⎢w⎦⎥ ⎡cosθ − sinθ Tx cosθ −Ty sinθ ⎤⎡ x⎤ = ⎢sinθ cosθ T sinθ +T cosθ ⎥⎢ y⎥ ⎢ x y ⎥⎢ ⎥ ⎣⎢ 0 0 1 ⎦⎥⎣⎢w⎦⎥ ¾ The computation of [ ] from [ ] [ ] is called matrix multiplication. The general form is: ⎡a b⎤⎡e f ⎤ ⎡ae + bg af + bh⎤ ⎢ ⎥⎢ ⎥ = ⎢ ⎥ ⎣c d⎦⎣g h⎦ ⎣ce + dg cf + dh⎦ ¾ A sequence of transformations can be lumped in a single matrix via matrix multiplications ¾ Scaling relative to a fixed point (m, n) 1. Translate the point(-m, -n) to the origin 2. Scale the object by Sx, Sy scale factors 3. Translate the point back (m, n) −1 []T = [Tt ] [S][Tt ] ⎡1 0 m⎤⎡Sx 0 0⎤⎡1 0 − m⎤ ⎢ ⎥⎢ ⎥⎢ ⎥ Column Vectors []T = ⎢0 1 n ⎥⎢ 0 Sy 0⎥⎢0 1 − n ⎥ ⎣⎢0 0 1 ⎦⎥⎣⎢ 0 0 1⎦⎥⎣⎢0 0 1 ⎦⎥ 9 ¾ Rotation about an arbitrary point (m, n) −1 []T = [Tt ][R][Tt ] ⎡ 1 0 0⎤⎡ cosθ sinθ 0⎤⎡1 0 0⎤ ⎢ ⎥⎢ ⎥⎢ ⎥ Row Vectors []T = ⎢ 0 1 0⎥⎢− sinθ cosθ 0⎥⎢ 0 1 0⎥ ⎣⎢− m − n 1⎦⎥⎣⎢ 0 0 1⎦⎥⎣⎢m n 1⎦⎥ P =[x y h] Rotation about an arbitrary point 1. Translate the point(-m, -n) to the origin P(m, n) 2. Rotate the object about the origin by θo 3. Translate back (m,n) −1 []T = []Tt [R][Tt ] ⎡ 1 0 0⎤⎡ cosθ sinθ 0⎤⎡1 0 0⎤ ⎢ ⎥⎢ ⎥⎢ ⎥ []T = ⎢ 0 1 0⎥⎢− sinθ cosθ 0⎥⎢ 0 1 0⎥ ⎣⎢− m − n 1⎦⎥⎣⎢ 0 0 1⎦⎥⎣⎢m n 1⎦⎥ Ex: Center of an object (4,3) , rotate it about the center by 90o CC ⎡ 0 1 0⎤ ⎡0 −1 7 ⎤⎡x⎤ ⎢ ⎥ ⎢ ⎥⎢ ⎥ []x y 1 ⎢−1 0 0⎥ OR ⎢1 0 −1⎥⎢y⎥ ⎣⎢ 7 −1 1⎦⎥ ⎣⎢0 0 1 ⎦⎥⎣⎢1 ⎦⎥ 10 y =mx+b Reflection O(3) Reflection is a special case of rotation SO(3) when angel of rotation is 180° about an axis in the plane b θ θ = tan −1(m) sin(180) = 0,cos(180) = −1 ⎡1 0 0⎤ ⎢ ⎥ []T = ⎢0 −1 0⎥ ⎣⎢0 0 1⎦⎥ y =mx+b Reflection about an arbitrary line 1. Translate (0, -b) so that the line passes through the origin 2. Rotate the line about the x axis by -θo b 3. Reflect object about the x axis θ 4. Rotate back the line by θo 5. Translate back (0,b) −1 −1 −1 θ = tan (m) []T = [][][]Tt Tr R [Tr ] [Tt ] ⎡1 0 0 ⎤⎡ 2/ 5 1/ 5 0⎤⎡1 0 0⎤⎡2/ 5 −1/ 5 0⎤⎡1 0 0⎤ Alternate []T = ⎢0 1 −2⎥⎢−1/ 5 2/ 5 0⎥⎢0 −1 0⎥⎢1/ 5 2/ 5 0⎥⎢0 1 2⎥ Method? ⎣0 0 1 ⎦⎣ 0 0 1⎦⎣0 0 1⎦⎣ 0 0 1⎦⎣0 0 1⎦ 1 Ex: Reflect triangle (2,4),(4,6),(2,6) about line y = 2 (x + 4) ⎡3/5 4/5 −8/5⎤⎡2 4 2⎤ ⎡14/5 28/5 22/5⎤ ⎢4/5 −3/5 16/5⎥⎢4 6 6⎥ = ⎢12/5 14/5 6/5 ⎥ ⎣ 0 0 1 ⎦⎣1 1 1⎦ ⎣ 1 1 1 ⎦ 11 Mid-point Transformation Consider a straight line between A(x ,y ) and B(x ,y ).