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 ). Let us try A* M’ B* 1 1 2 2 E to transform this line B using a general a 2×2 transformation matrix M ⎡a b⎤ [T ] = ⎢ ⎥ F’ ⎣c d⎦ A ⎡A*⎤ ⎡A⎤ ⎡x y ⎤⎡a b⎤ ⎡ ax + cy bx + dy ⎤ ⎡x* y* ⎤ = [T ] = 1 1 = 1 1 1 1 = 1 1 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ * * ⎥ ⎣B*⎦ ⎣B⎦ ⎣x2 y2 ⎦⎣c d⎦ ⎣ax2 + cy2 bx2 + dy2 ⎦ ⎣x2 y2 ⎦

* * * * * * ⎡ x1 + x2 y1 + y2 ⎤ ⎡ax1 + cy1 + ax2 + cy2 bx1 + dy1 + bx2 + dy2 ⎤ []xm ym = ⎢ ⎥ = ⎢ ⎥ ⎣ 2 2 ⎦ ⎣ 2 2 ⎦ ⎡ x + x y + y x + x y + y ⎤ = a 1 2 + c 1 2 b 1 2 + d 1 2 (1) ⎣⎢ 2 2 2 2 ⎦⎥

⎡ x1 + x2 y1 + y2 ⎤⎡a b⎤ ⎡ x1 + x2 y1 + y2 x1 + x2 y1 + y2 ⎤ []x'm y'm = ⎢ ⎥⎢ ⎥ = ⎢a + c b + d ⎥ (2) ⎣ 2 2 ⎦⎣c d⎦ ⎣ 2 2 2 2 ⎦

Mid-point Transformation From (1) and (2) we can conclude that by recursively continuing this process, we will cover all A* M’ B* E points on the line AB and hence their B images on A*B* by corresponding map. M

A F’ But a general proof of end points transformation actually transforms the whole line can be given usign the section formula. Any point m that divides the line AB in the ratio p:q can be given as: 1 m = ( pb + qa) p + q Needless to say that the same is applicable to A*B*. The special case of mid point transformation occurs when p:q=1.

12 Transformation of intersecting lines E’ When two intersecting lines are transformed using A’ B’ E general 2x2 transformation B matrix the resulting lines are also intersecting Consider 2 intersection lines F F’ y = m1x + b1 A ⎡ ⎤ y = m x + b b1 − b2 b1m2 − b2m1 2 2 [][X i = xi yi ]= ⎢ ⎥ The point of intersection: ⎣m2 − m1 m2 − m1 ⎦ ⎡a b⎤ ⇒ []X Transformed point of i ⎢c d⎥ intersection: ⎣ ⎦

' ' ⎡a(b1 − b2 ) + c(b1m2 − b2m1) b(b1 − b2 ) + d(b1m2 − b2m1)⎤ []xi yi = ⎢ ⎥ ⎣ m2 − m1 m2 − m1 ⎦

Transformation of intersecting lines E’ Observe these points A’ B’ In the last slide E 1. A pair of non-perpendicular lines B get transformed to perpendicular lines

F’ -1 F 2. By inverse T a pair of perpendicularA lines can get transformed to non-perpendicular lines 3. Such a result can have disastrous effect on the resulting geometry

13 Transformation of two parallel lines When two parallel lines are transformed using general 2x2 transformation matrix the resulting lines remain parallel Consider 2 lines parallel to each other between the end points A(x1,y1),B(x2,y2) and CD. Since they are parallel the slope is the same given by y − y m = 2 1 x2 − x1

⎡a c⎤⎡x1 x2 ⎤ ⎡ax1 + cy1 ax2 + cy2 ⎤ [][]T X = ⎢ ⎥⎢ ⎥ = ⎢ ⎥ ⎣b d⎦⎣y1 y2 ⎦ ⎣bx1 + dy1 bx2 + dy2 ⎦ (bx + dy ) − (bx + dy ) b(x − x ) + d(y − y ) m′ = 2 2 1 1 = 2 1 2 1 (ax2 + cy2 ) − (ax1 + cy1 ) a(x2 − x1 ) + c(y2 − y1 ) (y − y) b + d 2 (x − x ) b + dm m′ = 2 1 1 = (y − y ) a + cm a + c 2 1 (x2 − x1 )

Intersecting lines and rigid body transformations When do perpendicular lines transform as perpendicular lines? Consider the scalar(dot) and vector products of two vectors as follows

V1.V2 =V1xV2x +V1yV2 y = V1 V2 cosθ

v v V1×V2 =(V1xV2x −V1yV2 y )k = V1 V2 k sinθ Let us now transform these vectors by general 2x2 T and find dot and cross products again

⎡a c⎤⎡V1x V2x ⎤ ⎡aV1x + cV1y aV2x + cV2 y ⎤ ⎢ ⎥⎢ ⎥ = ⎢ ⎥ ⎣b d⎦⎣V1y V2x ⎦ ⎣bV1x + dV1y bV2x + dV2 y ⎦ 2 2 2 2 ′ ′ (a + b )V1xV2x + (c + d )V1yV2 y + (ac + bd )(V1xV2 y + V1yV2x ) = V1 V2 cos θ r r ′ ′ (ad − bc)(V1xV2 y − V2xV1y )k = V1 V2 k sin θ

We require for magnitude and angle to remain unchanged 2 2 2 2 −1 T a + b = c + d = 1; ac + bd = 0; ad − bc = +1 OR [T ][T ] = [T ][T ][]= I

14 Transformation of Unit square and area Observe unit square being scaled and sheared along x and y axes simultaneously

C’ D’ (a+c, b+d) (c,d) D C (a,b) B’ AB (o,0)A’ ⎡0 1 1 0⎤ ⎡a c⎤⎡0 1 1 0⎤ ⎡0 a a + c c⎤ P = ⎢ ⎥; and P′ = ⎢ ⎥⎢ ⎥ = ⎢ ⎥ ⎣0 0 1 1⎦ ⎣b d⎦⎣0 0 1 1⎦ ⎣0 b b + d d⎦ 1 1 Ap = (a + c)(b + d ) − 2 (ab) − 2 (cd )

− c (b+b+d )−b(c+a+c)=ad −bc 2 2

= det[T ]

∴ Ap = As det[T ]

Area Scaling example A triangle with vertices (1,0), (0,1) and (-1,0) is transformed by ⎡ 3 2⎤ [T ] = ⎢ ⎥ ⎣−1 2⎦

1 Area Ai = 2 (2)(1) =1 sq.units Vertices after transformation are ⎡ 1 0⎤ ⎡ 3 2 ⎤ ⎢ ⎥⎡ 3 2⎤ ⎢ ⎥ [X '] = ⎢ 0 1⎥⎢ ⎥ = ⎢−1 2 ⎥ ⎣−1 2⎦ ⎣⎢−1 0⎦⎥ ⎣⎢− 3 − 2⎦⎥

1 Transformed area,A t = Ai 2 (4)(4) = 8 sq.units

15 Geometrical Interpretation of Overall Scaling ⎡1 0 0⎤ ⎢ ⎥ [][][]x* y * h = x y 1 ⎢0 1 0⎥ = x y s ⎣⎢0 0 s⎦⎥ ⎡ x y ⎤ = 1 ⎣⎢ s s ⎦⎥

h<1

h=1

h>1

Viewing Transformations • It is a transformation from world CS to Screen CS. The dimensions of the WCS may be in any chosen system of units SI or FPS etc the latter is measured in pixels. 1. Translate the MCS base point to origin 2. Apply the scaling necessary to fit into the screen limits (xmin,ymin and xmax ymax) 3. Move the base point back to its original position

−1 [T ]= [Tt ][S][Tt ]

u −u ⎡ max min 0 0⎤ 1 0 0 xmax −xmin 1 0 0 ⎡ ⎤⎢ v −v ⎥⎡ ⎤ []T = 0 1 0 0 max min 0 0 1 2 ⎢ ⎥⎢ ymax − ymin ⎥⎢ ⎥ ⎣−xmin − ymin 1⎦ 0 0 1 ⎣umin vmin 1⎦ ⎣⎢ ⎦⎥

16 Translation

• What are invariant? – Lengths –Angles • Which transformation group? – Congruent • det[T]=1

Translation + Dilation

• What are invariant? – Ratio of Lengths –Angles • Which transformation group? – similarity • det[T]=1

17 Translation + Rotation

• What are the invariants? • Lengths • Angles • Which transformation group? • Congruent • det[T]=1

Reflection

• What are the invariants? • Lengths • Angles • Which transformation group? • Congruent • det[T] = -1

18 Reflection+Translation=Glide

• What are the invariants? • Lengths • Angles • Which transformation a group? m • Congruent a • det[T] = -1

• Glide= Rm+Ta=Ta+Rm

Shear Transformation

• What are invariant? • parallelism

4

2 • Which transformation 5 10 group? -2 • affine

-4

6

19 Projective Transformation

• What are invariant? • Incidence • Cross ratios of 4

lengths 2 • Order of curves 5 10

-2

-4

Isometries of a Square

• How many isometries? Cayle Table of isometries of a square

• identity i 90 180 270 H V D1 D2

• Rotational I i 90 180 270 H V D1 D2 • Reflection 90 90 180 270 i D1 D2 V H 180 180 270 i 90 V h D2 D1

i 90 180 270 270 270 i 90 180 D2 D1 H V

H H D2 V D1 i 180 270 90 h v D1 D2 V V D1 H D2 180 i 90 270

D1 D1 H D2 V 90 270 i 180

D2 D2 V D1 H 270 90 180 i

20 Transformations Possible in different Geometries

Transformation / Euclidian Similarity Affine Projective Geometries Rotation 3 3 3 3 Translation 3 3 3 3 Uniform Scaling U 3 3 3 Non-uniform Scaling U U 3 3 Shear U U 3 3 Central Projection U U U 3

Invariant Quantities in different Geometries

Transformation / Euclidian Similarity Affine Projective Geometries Lengths 3 U U U Angles 3 3 U U Ratios of Lengths 3 3 U U Parallelism 3 3 3 U Incidence 3 3 3 3 Cross-ratios of 3 3 3 3 Lengths

21