3D Transformations3D Transformations

Translation x’ = x + txy’ = y + tyz’ = z + tz

P = P’ =

T =

P’ = T . P

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

xyz1

x’y’z’1

x

y

P

P’

z

Scaling

S =

P’ = S . P

wrt a fixed point (xf, yf, zf) :

T(xf, yf, zf) . S(sx, sy, sz) . T(-xf, -yf, -zf)

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

x

y

z

Rotation z-axis rotationx’ = x.cos– y.siny’ = x.sin y.cosz’ = z

R =

P’ = R . P

cos -sin sin cos

x

y

z

Rotation x-axis rotationreplace x -> y -> z -> x in z-axis rotation y’ = y.cos– z.sinz’ = y.sin z.cosx’ = x

R =

y-axis rotationreplace x -> y -> z -> x in x-axis rotationz’ = z.cos– x.sinx’ = z.sin x.cosy’ = y

R =

1 0 0 0 0 cos -sin 0 0 sin cos 00 0 0 1

x

y

z

cos 0 -sin 0 0 1 0 0sin 0 cos 0

0 0 0 1

x

y

z

z-axis rotationx’ = x.cos– y.siny’ = x.sin y.cosz’ = z

Rotation

Rotation about an arbitrary axis parallel to a coordinate axis

P’ = T-1 . Rx() . T. Px

y

z

Rotation

x

y

z

Rotation about an arbitrary axis NOT parallel to a coordinate axis

Rotation

x

y

z

Rotation about an arbitrary axis NOT parallel to a coordinate axis

1. Translate so that rotation axis passes through the origin

Rotation

x

y

z

Rotation about an arbitrary axis NOT parallel to a coordinate axis

1. Translate so that rotation axis passes through the origin

2. Rotate so that rotation axis coincides with a coordinate axis

Rotation

x

y

z

Rotation about an arbitrary axis NOT parallel to a coordinate axis

1. Translate so that rotation axis passes through the origin

2. Rotate so that rotation axis coincides with a coordinate axis

3. Rotate

Rotation

x

y

z

Rotation about an arbitrary axis NOT parallel to a coordinate axis

1. Translate so that rotation axis passes through the origin

2. Rotate so that rotation axis coincides with a coordinate axis

3. Rotate

4. Inverse rotation

Rotation

x

y

z

Rotation about an arbitrary axis NOT parallel to a coordinate axis

1. Translate so that rotation axis passes through the origin

2. Rotate so that rotation axis coincides with a coordinate axis

3. Rotate

4. Inverse rotation

5. Inverse translation

Reflection

1800 rotation about x-axis

a combination of translation and rotation

y

x

z

x

z

y

Shear

y

x

z

y

x

z

1 0 shx –shx.zref0 1 shy –shy.zref0 0 1 0 0 0 0 1

OpenGL

glTranslate*(tx, ty, tz)• f (float)

• d (double)

glRotate* (theta, vx, vy, vz)• (vx, vy, vz) vector defines the orientation of the

rotation axis that passes through the coordinate origin

glScale*(sx, sy, sz)

OpenGL

glMatrixMode(GL_MODELVIEW)• sets up the matrix for transformations (4x4 modelview

matrix)

glLoadIdentity ( )• assigns identity matrix to the current matrix

glLoadMatrix*(16-element array)• assigns a 16-element array (in column major order) to the

current matrix glMultMatrix*(16-element array)• postmultiplies a 16-element array (M’) with the current

matrix (M) : M <- M.M’

OpenGL

glMatrixMode(GL_MODELVIEW);

glLoadIdentity ( );

glMultMatrixf(M2);

glMultMatrixf(M1);

/* M = M2 . M1 */

OpenGL

Matrix Stack

Initially stack contains identity matrix Maximum stack depth is 32 glGetIntegerv (GL_MAX_MODELVIEW_STACK_DEPTH, stacksize)• returns the number of positions available in the modelview stack

glGetIntegerv (GL_MODELVIEW_STACK_DEPTH, nummats)• returns the number of matrices currently in the stack

glPushMatrix()• copies the current matrix at the top of the stack

glPopMatrix()• destroys the matrix at the top of the stack

OpenGL glMatrixMode(GL_MODELVIEW);

glColor3f(0.0, 0.0, 1.0);Recti(50, 100, 200, 150);

glColor3f(1.0, 0.0, 0.0);glTranslatef(-200.0, -50.0, 0.0);Recti(50, 100, 200, 150);

glLoadIdentity ( ); glRotatef(90.0, 0.0, 0.0, 1.0);Recti(50, 100, 200, 150);

glLoadIdentity ( );glScalef(-0.5, 1.0, 1.0);Recti(50, 100, 200, 150);

OpenGL glMatrixMode(GL_MODELVIEW);

glColor3f(0.0, 0.0, 1.0);Recti(50, 100, 200, 150);

glPushMatrix();glColor3f(1.0, 0.0, 0.0);glTranslatef(-200.0, -50.0, 0.0);Recti(50, 100, 200, 150);

glPopMatrix();glPushMatrix();

glRotatef(90.0, 0.0, 0.0, 1.0);Recti(50, 100, 200, 150);

glPopMatrix();glScalef(-0.5, 1.0, 1.0);Recti(50, 100, 200, 150);