Fundamental Concepts of Computer Graphics: Vectors and Transformations

1. Computer GraphicsLecture 112D Transformations ITaqdees A. Siddiqics602@vu.edu.pk

2. Vectors • Another important mathematical concept used in graphics is the Vector • If P1 = (x1, y1, z1) is the starting point and P2 = (x2, y2, z2) is the ending point, then the vector V = (x2 – x1, y2 – y1, z2 – z1)

3. and if P2 = (x2, y2, z2) is the starting point and P1 = (x1, y1, z1) is the ending point, then the vector V = (x1 – x2, y1 – y2, z1 – z2)

4. This just defines length and direction, but not position

5. y (5, 2) v x v' (5, 0) Vector Projections • Projection of v onto the x-axis

6. y (5, 3, 2) v x v' (5, 0, 2) z • Projection of v onto the xz plane

7. y |v| (5, 2) a x 2D Magnitude and Direction • The magnitude (length) of a vector: |V| = sqrt( Vx2 + Vy2 ) • Derived from the Pythagorean theorem • The direction of a vector: a = tan-1 (Vy / Vx) • a is angular displacementfrom the x-axis

8. 3D Magnitude and Direction • 3D magnitude is a simple extension of 2D |V| = sqrt( Vx2 + Vy2 + Vz2 ) • 3D direction is a bit harder than in 2D • Need 2 angles to fully describe direction • Latitude/longitude is a real-world example

9. Direction Cosines are often used: • a, b, and g are the positive angles that the vector makes with each of the positive coordinate axes x, y, and z, respectively cos a = Vx / |V| cos b = Vy / |V| cos g = Vz / |V|

10. Vector Normalization • “Normalizing” a vector means shrinking or stretching it so its magnitude is 1 • creating unit vectors • Note that this does not change the direction • Normalize by dividing by its magnitude; V = (1, 2, 3) |V| = sqrt( 12 + 22 + 32) = sqrt(14) = 3.74

11. Vector Normalization Vnorm = V / |V| = (1, 2, 3) / 3.74 = (1 / 3.74, 2 / 3.74, 3 / 3.74) = (.27, .53, .80) |Vnorm| = sqrt( .272 + .532 + .802) = sqrt( .9 ) = .95 • Note that the last calculation doesn’t come out to exactly 1. This is because of the error introduced by using only 2 decimal places in the calculations above.

12. Vector Addition • Equation: V3 = V1 + V2 = (V1x+V2x , V1y+V2y , V1z+V2z) • Visually: y y V2 V3 V2 V1 V1 x x

13. y y V3 V2 V2 V1 V1 x x Vector Subtraction • Equation: V3 = V1 - V2 = (V1x-V2x, V1y-V2y, V1z-V2z) • Visually:

14. Dot Product • The dot product of 2 vectors is a scalar V1. V2 = (V1x V2x) + (V1y V2y )+ (V1z V2z ) • Or, perhaps more importantly for graphics: V1. V2 = |V1| |V2| cos(q) where q is the angle between the 2 vectors and q is in the range 0 q y V2 q V1 x

15. y V2 V1 x • Why is dot product important for graphics? • It is zero iff the 2 vectors are perpendicular • cos(90) = 0

16. The Dot Product computation can be simplified when it is known that the vectors are unit vectors V1. V2 = cos(q) because |V1| and |V2| are both 1 • Saves 6 squares, 4 additions, and 2 square roots

17. Cross Product • The cross product of 2 vectors is a vector V1x V2 = ( V1y V2z - V1z V2y , V1z V2x - V1x V2z , V1x V2y - V1y V2x ) • Note that if you are into linear algebra there is also a way to do the cross product calculation using matrices and determinants

18. Again, just as with the dot product, there is a more graphical definition: V1x V2 = u |V1| |V2| sin(q) where q is the angle between the 2 vectors and q is in the range 0 q and u is the unit vector that is perpendicular to both vectors • Why u? • |V1| |V2| sin(q) produces a scalar and the result needs to be a vector

19. V2 u V1 • The direction of u is determined by the right hand rule • The perpendicular definition leaves an ambiguity in terms of the direction of u • Note that you can’t take the cross product of 2 vectors that are parallel to each other • sin(0) = sin(180) = 0  produces the vector (0, 0, 0)

20. Forming Coordinate Systems • Cross products are great for forming coordinate system frames (3 vectors that are perpendicular to each other) from 2 random vectors • Cross V1 and V2 to form V3 • V3 is now perpendicular to both V1 and V2 • Cross V2 and V3 to form V4 • V4 is now perpendicular to both V2 and V3 • Then V2, V4, and V3 form your new frame

21. V1 and V2 are in the new xy plane V2 V2 V1 V3 V1 V2 x V3 z V1 V1 V4 y

22. Basic Transformations • Translation • Rotation • Scaling

23. Translation • Straight line movement • Rotation • Circular movement w.r.t pivot • Scaling • Change of size

24. 6 5 4 Y 3 2 1 0 1 2 3 4 5 6 7 8 9 10 X Translation tx = 2 ty = 3

25. Translation Distances tx, ty • For point P(x,y) after translation we have P′(x′,y′) • x′ = x + tx , y′ = y + ty • (tx, ty) is Translation vector

26. Now x′ = x + tx , y′ = y + ty can be expressed as a single matrix equation: P′ = P + T Where P = P′ = T =

27. Rigid Body Transformation • Objects are moved without deformation • Every point on the object is translated by the same amount • Typically all the endpoints are translated and object is redrawn using new endpoint positions

28. Rotation Y 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 10 X

29. Repositions an object along a circular path in xy-plane • Rotation Angle θ • Rotation Point (xr, yr) • θ is +ve  counterclockwise rotation • θ is -ve  clockwise rotation • θ is zero  ?

30. For simplicity, consider the pivot at origin and rotate point P (x,y)where x = r cosФ and y = r sinФ • If rotated by θ then: • x′ = r cos(Ф + θ) = r cosФ cosθ – r sinФ sinθand • y′ = r sin(Ф + θ) = r cosФ sinθ + r sinФ cosθ

31. Replacing r cosФ with x and r sinФ with y, we have: • x′ = x cosθ – y sinθand • y′ = x sinθ + y cosθ

32. Column vector representation: P′ = R . P • Where

33. For Row vector representation of P and P′, in order to get the same results: • Which is actually transpose of original R, therefore: P′T = (R . P)T = PT . RT

34. When rotating about (xr,yr) • x’ = xr + (x - xr) cosθ – (y - yr) sinθ • y’ = yr + (x - xr) sinθ – (y - yr) cosθ • A better way of expressing these as matrix operations would be discussed later

35. Rigid Body Transformation • Objects are rotated without deformation • Every point on the object is rotated by the same angle • Typically all the endpoints are rotated and object is redrawn using new endpoint positions

36. 6 5 4 Y 3 2 1 0 1 2 3 4 5 6 7 8 9 10 X Scaling

37. Scaling • Scaling alters the size of an object • Scaling factors Sx and Sy are used • For polygons, coordinates of each vertex may be multiplied by Sx & Sy to produce the transformed coordinates: x′ = x.Sx y′ = y.Sy

38. In matrix form it can be expressed as : P′ = S.P

39. Scaling factor > 1 Scaling factor < 1 Scaling factor = 1 Same valued scaling factors  uniform scaling in x and y Different valued scaling factors  differential scaling in x and y

40. When scaling w.r.t. origin, both the size and distance form origin get scaled. Size and distance fro origin: for Scaling factor 2 (double the size and double the distance from origin.) for Scaling factor 0.5 (reduce the size to 50% and also the distance from origin to 50%.

41. This can be controlled by scaling w.r.t. a fixed point (xf,yf) • x’ = xf + (x - xf)Sx • y’ = yf + (y - yf)Sy • These can be rewritten as: • x’ = x. Sx + xf (1 – Sx) • y’ = y. Sy + yf (1 – Sy) • Where the terms xf (1 – Sx) and yf (1 – Sy) are constant for all points in the object

42. Computer GraphicsLecture 112D Transformations ITaqdees A. Siddiqics602@vu.edu.pk