1. 3D Coordinate Systems and Transformations Revision 1
3D Coordinate Systems and Transformations Revision 1.1 Copyright Zachary Wartell, University of North Carolina at Charlotte, 2006 All Rights Reserved
Textbook: Chapter 5
©Zachary Wartell
4/17/2017

2. 3D Coordinate Systems
©Zachary Wartell
, Larry F. Hodges
4/17/2017

3. Points & Vectors
3D affine space: points (locations), vectors (alignment with signed magnitude)
operations:
Points: pi - pj = v, pi + v = pj (p + 0 = p)
Vectors: vi ±vj = vk , vj = a ∙ vk (a  R)
length(v), |v| =
dot product: vivj =
©Zachary Wartell
4/17/2017

4. Points & Vectors (cont.)
operations:
vectors:
cross product: vi×vj = vk
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4/17/2017

5. Parametric Definition of a Line
Given two points p1 = (x1, y1, z1), p2 = ( x2, y2, z2)
x = x1 + t (x2 - x1)
y = y1 + t (y2 - y1)
z = z1 + t (z2 - z1)
t > 1
t = 1
t = 0 p2
t < 0 p1
Given a point p1 and a vector v = (xv, yv, zv)
x = x1 + t xv, y = y1 + t yv , z = z1 + t zv
COMPACT FORM: p = p1 + t [p2 - p1] or p = p1 + vt
©Zachary Wartell
, Larry F. Hodges
4/17/2017

6. Equation of a plane Ax + By + Cz + D = 0
Alternate Form: A'x + B'y + C'z +D' = 0
where A' = A/d, B' = B/d, C' = C/d, D' = D/d
d =
Distance between a point and the plane is given by
A'x + B'y + C'z +D' (sign indicates which side)
Given Ax + By + Cz + D = 0 Then (A, B, C) is a normal vector
Pf: Given two points p1 and p2 in the plane, the vector (p2 - p1) is in the plane and (A,B,C) • (p2 - p1) = (Ax2 + By2 +Cz2) - (Ax1 + B y1 + Cz1) = ( - D ) ( - D ) = 0
©Zachary Wartell
, Larry F. Hodges
4/17/2017

7. Derivation of Plane Equation
To derive equation of the plane given three points: p1, p2 , p3
(p3 - p1) x (p2 - p1) = n, orthogonal vector
Given a general point p = (x,y,z)
n • [p - p1] = 0 if p is in the plane. p
Or given a point (x,y,z) in the plane and normal vector n then n • (x,y,z) = -D
©Zachary Wartell
, Larry F. Hodges
4/17/2017

8. Basic Transformations
Translation
Scale
Rotation
Shear
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9. Translation
Point or vector
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10. Scale (about origin)
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11. Rotation (about coordinate axis)
Positive Rotations are defined as follows
Axis of rotation is Direction of positive rotation is
x y to z
y z to x
z x to y
©Zachary Wartell
, Larry F. Hodges
4/17/2017

12. Rotations
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13. Shears XY sheared on Z y x z
Of course other combinations occur too, e.g. ZX sheared on Y, X sheared on Y,
etc.
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4/17/2017

14. Rotation About An Arbitrary Axisp2 u Z θ p1 Y X
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15. Rotation About An Arbitrary Axis
Translate p1 to origin p2
T(-p1) u Z p1 u' Y X
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4/17/2017

16. Rotation About An Arbitrary Axis
Rotate u' to u'' in YZ plane (rotation about Y axis, Ry(-β))
Ry(-β)) ∙ T(-p1) Z u' uz c Y a ß uy b ux X
©Zachary Wartell
, Larry F. Hodges
4/17/2017

17. Rotation About An Arbitrary Axis
Rotate u' to u'' in YZ plane (rotation about Y axis, Ry(-β))
Ry(-β)) ∙ T(-p1) Z
u'' u' uz c Y a ß uy b ux X
©Zachary Wartell
, Larry F. Hodges
4/17/2017

18. Rotation About An Arbitrary Axis
After Ry(-ß), angle α and u'' lies in the y-z plane.
Next, Rotate u'' to Z axis (u''') (rotation about X axis, Rx(α))
Rx(α) ∙ Ry(-β) ∙ T(-p1)
u'' Z a c uz Y α uy ux X
©Zachary Wartell
, Larry F. Hodges
4/17/2017

19. Rotation About An Arbitrary Axis
After Rx(α), u'' lies in the Z axis as u'''.
Next, rotate about Z by θ (Rz(θ))
Rz(θ) ∙ Rx(α) ∙ Ry(-β) ∙ T(-p1)
u''' Z
u'' Y θ X
©Zachary Wartell
4/17/2017

20. Rotation About An Arbitrary Axis
Now we’ve rotated about U.
Next, apply the inverse transformations to place u''' back on u p2
R((p1,p2),θ) = T(p1) ∙ Ry(β) ∙ Rx(-α) ∙ Rz(θ) ∙ Rx(α) ∙ Ry(-β) ∙ T(-p1) u Z θ p1 Y X
©Zachary Wartell
4/17/2017

21. Rotation About An Arbitrary Axis
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4/17/2017

22. Rotation Representations So Far….
Euler angles
-Rx  Ry  Rz - note order is arbitrary, 6 possible
-3 real numbers
-interpolation problems
-gimbal lock
-compose rotation matrix: 27 *’s and 18 +’s
Matrix
-9 real numbers
-no good way to interpolate rotations
Axis-Angle
-4 real numbers
-again unclear how to interpolate rotations
-no clear way to compose rotations (must convert to
matrix)
©Zachary Wartell
4/17/2017

23. Euler Angles: Gimbal Lock
 Occurs when two rotation axes are aligned by
an intermediate rotation
 Example: Assume Euler order Rz ∙ Ry ∙ Rx
If Ry = 90 degrees, then
x-axis aligns with y-axis and Rz = -Rx Z Y X'
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4/17/2017 X

24. Rotation Matrix: Interpolation?
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4/17/2017

25. Axis-Angle: Composition
R(?,?) = R(uk,θk) ∙ R(uk-1,θk-1) ….. R(u1,θ1)
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4/17/2017

26. Quaternions Alternative rotation representation: 4 real numbers
interpolation between quaternions well defined
(“slerp”)
no gimbal lock
composition of quaternions: 16 *’s, 9+’s
(matrix 27 *’s and 18 +’s)
©Zachary Wartell
4/17/2017

27. Quaternions Rotation representation:
Relation to axis-angle representation (u,θ) is: y u θ x Z
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4/17/2017

28. Quaternion Rotation: Basic Idea
To rotate point p:
represent p as quaternion:
compute rotated quaternion point representation
extract standard coordinate representation p' from qp' y u p' p x Z
©Zachary Wartell
4/17/2017

29. Details of Rotation How do you multiply quaternions:
How do you find inverse of quaternion:
For rotations we assume unit quaternions, hence:
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4/17/2017

30. Quaternion Rotation: Complete Equation
To rotate point p: y u p' p x Z
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31. Quaternion Rotation Examples:
Recall:
We have examples:
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4/17/2017

32. How do quaternions help?
Quaternion composition: requires 16 *’s, 12+’s while 3x3 matrix composition needs 27*’s and 18+’s
No gimbal lock
interpolation between two orientations using “spherical linear interpolation” (slerp) of their quaternions: -constant angular velocity
-unique interpolation
-interpolation that is same regardless of coordinate
system used for computation
©Zachary Wartell
4/17/2017