361-07: 3d transformations - computing science - … unit vectors, i.e, orthonormal • property 2:...

3D Transformations

CMPT 361Introduction to Computer Graphics

Torsten Möller

Schedule• Geometry basics• Affine transformations• Use of homogeneous coordinates• Concatenation of transformations• 3D transformations• Transformation of coordinate systems• Transform the transforms• Transformations in OpenGL

Transformations in 3D• Add a z-axis to (x, y) plane

– right-handed system:• positive z pointing towards us• positive rotation counter-clockwise• Standard math convention (used in our presentation

and OpenGL)– left-handed system:

• positive z pointing away from us• positive rotation clockwise• Used in some graphics systems

(z-axis as depth), e.g., POV-Ray, and Renderman

Translation in 3D• Again, we use homogeneous coordinates

T (tx, ty, tz) =

⇧⇧⇤

1 0 0 tx0 1 0 ty0 0 1 tz0 0 0 1

⌃⌃⌅

Scaling in 3D

S(sx, sy, sz) =

⇧⇧⇤

sx 0 0 00 sy 0 00 0 sz 00 0 0 1

⌃⌃⌅

Rotation in 3D• Around z-axis

• Around x-axis

• Around y-axis

Rz(�) =

⇧⇧⇤

cos � � sin� 0 0sin� cos � 0 0

0 0 1 00 0 0 1

⌃⌃⌅

Rx(�) =

⇧⇧⇤

1 0 0 00 cos � � sin� 00 sin� cos � 00 0 0 1

⌃⌃⌅

Ry(�) =

⇧⇧⇤

cos � 0 sin� 00 1 0 0

� sin� 0 cos � 00 0 0 1

⌃⌃⌅

• Property 1: columns and rows are mutually orthogonal unit vectors, i.e, orthonormal

• Property 2: determinant of M = 1

• product of any pair of orthonormal matrices is also orthonormal

• orthonormality: inverse = transpose (PT= P−1)

Properties of rotation matrix

⇧⇧⇤

r11 r12 r13 0r21 r22 r23 0r31 r32 r33 00 0 0 1

⌃⌃⌅

Another nice property• row vectors: unit vectors which rotate into

principal axes, i.e., [1 0 0]T, [0 1 0]T, and [0 0 1]T

• column vectors: unit vectors into which principle axes rotate (obviously)

⇤100

⇤r11 r12 r13

r21 r22 r23

r31 r32 r33

⇤r11

⇤010

⇤r11 r12 r13

r21 r22 r23

r31 r32 r33

⇤r21

⇤001

⇤r11 r12 r13

r21 r22 r23

r31 r32 r33

⇤r31

Shearing in 3D• In (y, z) w.r.t. x value

• In (z, x) w.r.t. y value

• In (x, y) w.r.t. z value

SHyz =

⇧⇧⇤

1 0 0 0shy 1 0 0shz 0 1 00 0 0 1

⌃⌃⌅

SHxz =

⇧⇧⇤

1 shx 0 00 1 0 00 shz 1 00 0 0 1

⌃⌃⌅

SHxy =

⇧⇧⇤

1 0 shx 00 1 shy 00 0 1 00 0 0 1

⌃⌃⌅

Inverse Transforms• Translation: negate tx, ty, tz

• Scaling: change sx to 1/sx , etc.• Rotation: negate the angle• Shearing: negate shy, shz, etc.

General 3D transformations• Any arbitrary sequence of rotation, translation

scaling, and shear can be represented as:

• where upper left 3 × 3 is the combined scaling, rotation, and shearing; [tx ty tz]T for translation

⇧⇧⇤

r11 r12 r13 txr21 r22 r23 tyr31 r32 r33 tz0 0 0 1

⌃⌃⌅

Compound transforms• Just like in 2D, however …• Rotation is about more than one axis

• How should we do this?

P3 on (y, z) plane

Compute compound transform• Use of right-hand CS

• Translation by P1 so that P1 is at origin

• Rotation about y by (θ − 90°)to get P1P2 onto the (y, z) plane

T(–x1, –y1, –z1)

Compute compound transform• Rotation about x by φ to

get P1P2 to align with the positive z-axis

• Rotation about z by α to get P1P3 onto the (y, z) plane

Ry(θ − 90°)

Rx(φ)

Compute compound transform• Combined transformation:

Rz(α)

Rz(�)⇥Rx(⇤)⇥Ry(⇥ � 90�)⇥ T (�P1)

Alternative composition• Recall the “nice” properties of the rotation

matrices:

• Ri’s are the unit row vectors which rotate into principal coordinate axes, e.g., RRxT = [1 0 0]T

• Let us try to construct these directly, assuming the translation T(−P1) is already done.

Alternative composition• the unit vector to move to lie on the positive z

axis is:

• the unit vector that rotates into x is normal to the plane P1P2P3.

Alternative composition• By definition, Rz × Rx must rotate into the

remaining y-axis and:

• We are done:

M =�

R 00 1

⇥⇥ T (�P1),where R =

Exercise• How to get the jet into the desired direction of

flight (DOF)?

Special transformations• Points: we have been doing this so far• Lines: just transform the endpoint of a line• Planes: trickier

– if defined by 3 points, can transform points, – but ... more often defined by a plane equation

• By homogeneous coordinates we can write:

• with P = [x y z 1]T:

• Now, suppose we want to transform our space by matrix M

• To maintain NTP = 0 , we must also transform N. Let this transform be Q.

Plane transform

N =�

A B C D⇥T

NT P = 0

Plane transform: derivation• After the transform we have:

• and we would like to have:

• now some algebra:

Pn = MPNn = QN

NTn Pn = 0

NTn Pn = (QN)T (MP )

= NT (QT M)P= 0

Plane transform: result• This will hold when:

• hence:

QT M = kI

Q = M�T

Transformation of CS• So far: transform points on one

object with respect to the same coordinate system (CS)

• Sometimes need to change CS• e.g. we may have many objects,

each in its own CS, and we want to express all of them in some GLOBAL CS

CS1: P=P(6,5) CS2: P=P(2,3)

Transformation of CS• P(i) = point in coordinate system i• M2←1 converts representation of point in CS1 to

representation of point in CS2

• Alternate interpretation:M2←1 transforms axesof CS2 into axes of CS1

• By definition:• Hence:• Therefore:

• with other words, M2←1 transforms CS2 into CS1

DerivationP (2) = M2�1P

CS2P(2) = CS1P

CS2M2�1 = CS1

Transform of CS: example• Example: M2←1 = T(−4, −2), this is seen by

inspection• (2,3)T = T(−4, −2)(6,5)T

• CS1 = CS2 T(−4, −2)

CS1: P=P(6,5) CS2: P=P(2,3)

• Observe transitivity of this operator:– Given

– then

– so that

• this is/was our basic concatenation of transformations

Transform of CS: transitivity

P (j) = Mj�iP(i)

P (k) = Mk�jP(j)

P (k) = Mk�jMj�iP(i)

Mk�i = Mk�jMj�i

Transformation of CS• Example: what is M2←3?

– M2←3 = M2←3’ M3’←3 with CS3’ aligned with CS3 but having the same scale as CS2

– M3’←3 transforms CS3’ to CS3: S(0.5, 0.5)

– M2←3’ transforms CS2 to CS3’:T(2, 3)

– M2←3 = T(2, 3)S(0.5, 0.5)– Verify: – Alternative: M2←3 = S(0.5, 0.5) T(4, 6)

CS3’

• What is M3 ← 4?– M3←4 = M3←4’ M4’←4

with CS4’ as shown – M4’←4 transforms CS4’

to CS4: R(+45)– M3←4’ transforms CS3

to CS4’ : T(6.7, 1.8)– M3←4 = T(6.7, 1.8)R(+45)– Verify:

Transform of CS: example

Transforming the transforms• Suppose Qj is a transformation in CSj

• Need Qi that acts on points, with respect to CSi, just like Qj would on the same points

• Assume that we know Mi ← j

Qi?Pi = Mi�jPj

P ⇥i = Mi�jP

P ⇥j = QjPj

P ⇤i = Mi⇥jP

= Mi⇥jQjPj

= [Mi⇥jQjM�1i⇥j ]Pi

Qi = Mi⇥jQjM�1i⇥j

Example: How does a point P on the front tricycle wheel move in the world CS (wo) when the wheel rotates forward by an angle of α about its own zwh?

Transforming the transforms

Transforming the transformsP ⇥(wo) = Mwo�whP ⇥(wh)

= Mwo�whMwh�wh�P ⇥(wh�)

= Mwo�whT (�r, 0, 0)P ⇥(wh�)

= Mwo�whT (�r, 0, 0)Rz(��)P (wh)

αrP: original point

P’: transformed point

Coordinate Systems• The units in points are determined by the application and

are called– object (or model) coordinates– world coordinates

• Viewing specifications usually are also in object coordinates• transformed through

– eye (or camera) coordinates– clip coordinates– normalized device coordinates– window (or screen) coordinates

• OpenGL also uses some internal representations that usually are not visible to the application but are important in the shaders 37

model view transform

projection transform

CTM in OpenGL • OpenGL had a model-view and a projection

matrix in the pipeline which were concatenated together to form the CTM

• Angel emulates this process

Rotation, Translation, Scaling

mat4 r = Rotate(theta, vx, vy, vz)m = m*r;

mat4 s = Scale( sx, sy, sz)mat4 t = Transalate(dx, dy, dz);m = m*s*t;

mat4 m = Identity();

Create an identity matrix:

Multiply on right by rotation matrix of theta in degrees where (vx, vy, vz) define axis of rotation

Do same with translation and scaling:

Rotation, Translation, Scaling• Create an identity matrix:

• Multiply on right by rotation matrix of theta in degrees where (vx, vy, vz) define axis of rotation

• Do same with translation and scaling:

mat4 r = Rotate(theta, vx, vy, vz)m = m*r;

mat4 s = Scale( sx, sy, sz)mat4 t = Transalate(dx, dy, dz);m = m*s*t;

mat4 m = Identity();

Example• Rotation about z axis by 30 degrees with a

fixed point of (1.0, 2.0, 3.0)

• Remember that last matrix specified in the program is the first applied

mat4 m = Identity();m = Translate(1.0, 2.0, 3.0)* Rotate(30.0, 0.0, 0.0, 1.0)* Translate(-1.0, -2.0, -3.0);

Arbitrary Matrices• Can load and multiply by matrices defined in

the application program• Matrices are stored as one dimensional array of

16 elements which are the components of the desired 4 x 4 matrix stored by columns

• OpenGL functions that have matrices as parameters allow the application to send the matrix or its transpose

361-07: 3d transformations - computing science - … unit vectors, i.e, orthonormal • property 2:...

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