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## Transformations With OpenGL

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Object Representation

Objects are represented as a closed mesh of polygons

The finer the mesh the more detail can be captured

OpenGL

- To create a red polygon with 4 vertices:
- glBegindefines a geometric primitive:
- GL_POINTS, GL_LINES, GL_LINE_LOOP, GL_TRIANGLES, GL_QUADS, GL_POLYGON…
- All vertices are 3D and defined using glVertex*

glColor3f(1.0, 0.0, 0.0);

glBegin(GL_POLYGON);

glVertex3f(0.0, 0.0, 3.0);

glVertex3f(1.0, 0.0, 3.0);

glVertex3f(1.0, 1.0, 3.0);

glVertex3f(0.0, 1.0, 3.0);

glEnd();

OpenGL

- We can use per-vertex information.
- To create the RG colour square:

glShadeModel(GL_SMOOTH);

glBegin(GL_POLYGON);

glColor3f(1.0, 0.0, 0.0); // Red

glVertex3f(0.0, 0.0, 3.0);

glColor3f(0.0, 0.0, 0.0); // Black

glVertex3f(1.0, 0.0, 3.0);

glColor3f(0.0, 1.0, 0.0); // Green

glVertex3f(1.0, 1.0, 3.0);

glColor3f(1.0, 1.0, 0.0); // Yellow

glVertex3f(0.0, 1.0, 3.0);

glEnd();

Geometric Transformations

- Many geometric transformations are linear and can be represented as a matrix multiplication.
- Function f is linear iff:
- Implications: to transform a line we transform the end-points. Points between are affine combinations of the transformed endpoints.
- Given line defined by points P and Q, points along transformed line are affine combinations of transformed P and Q

Homogeneous Co-ordinates

- Basis of the homogeneous co-ordinate system is the set of n basis vectors and the origin position:
- All points and vectors are therefore compactly represented using their ordinates:

Co-ordinate Frame

- A Frame is a point P0 (the origin) and a set of vectors vn which define the basis for the vector space (the axes).
- Vectors of the frame take the form:
- Points defined using the frame are given as:
- Vectors are uniquely defined using the ordinatesan but points require extra information (i.e. the origin)
- In computer graphics we use homogeneous co-ordinates to express vector and affine quantities and transformations.

Homogeneous Co-ordinates

- Vectors have no positional information and are represented using ao = 0 whereas points are represented with ao = 1:
- Examples:

Points

Associated vectors

Scale

- all vectors are scaled from the origin:

Original

scale all axes

scale Y axis

offset from origin

distance from origin also scales

Rotation

- Rotations are anti-clockwise about the origin:

rotation of 45o about the Z axis

offset from origin rotation

Rotation

- 2D rotation of q about origin:
- 3D homogeneous rotations:
- Note:
- If M-1= MT then M is orthonormal. All orthonormal matrices are rotations about the origin.

Shear

- We shear along an axis according to another axis
- Shearing along X axis preserves y and z values.
- Shearing along Y axis preserves x and z values
- Shearing along Z axis preserves x and y values
- Point are stretched along the shear axis in proportion to the distance of the point along another axis.
- Example: shearing along X according to Y

Translation

- Translation only applies to points, we never translate vectors.
- Remember: points have homogeneous co-ordinate w = 1

translate along y

Affine Transformations

- All affine transformations are combinations of rotations, scaling and translations.
- Shear = rotation followed by a scale:
- Affine transformations preserve
- Collinearity
- Ratios of distances along a line (therefore parallelism)

Transformation Composition

- More complex transformations can be created by concatenating or composing individual transformations together.
- Matrix multiplication is non-commutative order is vital
- We can create an affine transformation representing rotation about a point PR:
- translate to origin, rotate about origin, translate back to original location

Transformation Composition

Rotation in XY plane by q degrees anti-clockwise about point P

Euler Angles

- Euler angles represent the angles of rotation about the co-ordinate axes required to achieve a given orientation (qx, qy, qz)
- The resulting matrix is:
- Any required rotation may be described (though not uniquely) as a composition of 3 rotations about the coordinate axes.
- Remember rotation does not commute order is important
- A frequent requirement is to determine the matrix for rotation about an given axis.
- Such rotations have 3 degrees of freedom (DOF):
- 2 for spherical angles specifying axis orientation
- 1 for twist about the rotation axis

Rotational DOF

Sometimes known as roll, pitch and yaw

Rotation about an axis

- Assume axis is defined by points P and Q therefore pivot point is P and rotation axis vector is:
- First we translate the pivot point to the origin T(-P)
- Now we determine a series of rotations to achieve the required rotation about the desired vector.
- This is conceptually simpler if we first rotate the axis and object so that the axis lines up with z say R(qy)R(qx)
- Now we rotate about z by the required angle q R(q)

Rotation about an axis

- Then we undo the first 2 rotations to bring us back to the original orientation R(-qx)R(-qy)
- Finally we translate back to the original position T(P)
- The final rotation matrix is:
- We need to determine Euler angles qx and qy which will orient the rotation axis along the z axis.
- We determine these using simple trigonometry.

Aligning axis with z

- Note that as shown the rotation about the x axis is anti-clockwise but the y axis rotation is clockwise.
- Therefore the required y axis rotation is -qy

Spherical Co-ordinates

- The set of all normal vectors define a unit sphere which is usually used to encode the set of all directions.
- Each normal vector now has only 2 degrees of freedom usually denoted using spherical co-ordinates (angles) = (q, f)

Normal Vectors

- It is frequently useful to determine the relationship between the spherical co-ordinate and the vector:

Transformations and OpenGL®

- glRotatef(angle, vx, vy, vz)
- rotates about axis (vx, vy, vz) by angle (specified in degrees)
- glTranslate(dx, dy, dz)
- translates by displacement vector (dx, dy, dz)
- glScalef(sx, sy, sz)
- apply scaling of sx in x direction, sy in y direction and sz in z direction (note that these values specify the diagonal of a required matrix)
- glLoadIdentity()
- creates an identity matrix (used for clearing all transformations)
- glLoadMatrixf(matrixptr)
- loads a user specified transformation matrix where matrixptr is defined as GLfloat matrixptr[16];

Transformations and OpenGL®

- OpenGL defines 3 matrices for manipulation of 3D scenes:
- GL_MODELVIEW: manipulates the view and models simultaneously
- GL_PROJECTION: performs 3D 2D projection for display
- GL_TEXTURE: for manipulating textures prior to mapping on objects
- Each acts as a state parameter; once set it remains until altered.
- Having defined a GL_MODELVIEW matrix, all subsequent vertices are created with the specified transformation.
- Matrix transformation operations apply to the currently selected system matrix:
- use glMatrixMode(GL_MODELVIEW) to select modeling matrix

Transformations and OpenGL®

Vertex Geometry Pipeline

MODELVIEW

matrix

PROJECTION

matrix

perspective

division

viewport

transformation

original

vertex

final window

coordinates

normalised device

coordinates

(foreshortened)

2d projection

of vertex onto

viewing plane

vertex in the

eye coordinate

space

Transformations and OpenGL®

- The MODELVIEW matrix is a 4x4 affine transformation matrix and therefore has 12 degrees of freedom:
- The MODELVIEW matrix is used for both the model and the camera transformation
- rotating the model is equivalent to rotating the camera in the opposite direction OpenGL uses the same transformation matrix
- this sometimes causes confusion!

Transformations and OpenGL®

- Each time an OpenGL transformation M is called the current MODELVIEW matrix C is altered:

glTranslatef(1.5, 0.0, 0.0);

glRotatef(45.0, 0.0, 0.0, 1.0);

Transformations and OpenGL®

- Transformations are applied in the order specified (with respect to the vertex) which appears to be in reverse:

glMatrixMode(GL_MODELVIEW);

glLoadIdentity();

glTranslatef(1.5, 0.0, 0.0);

glRotatef(45.0, 0.0, 0.0, 1.0);

glVertex3f(1.0, 0.0, 0.0);

original

rotate

translate

Camera & Model Transformations

- Given that camera and model transformations are specified using a single matrix we must consider the effect of these transformations on the coordinate frames of the camera and models.
- Assume we wish to orbit an object at a fixed orientation:
- translate object away from camera
- rotate around X to look at top of object
- then pivot around object’s Y in order to orbit properly.

Camera & Model Transformations

- If the GL_MODELVIEW matrix is an identity matrix then the camera frame and the model frame are the same.
- i.e. they are specified using the same co-ordinate system
- If we issue command glTranslatef(0.0, 0.0, -10.0) and then create the model:
- a vertex at [0, 0, 0] in the model will be at [0, 0, -10] in the camera frame
- i.e. we have moved the object away from the camera
- This may be viewed conceptually in 2 ways:
- we have positioned the object with respect to the world frame
- we have moved the world-frame with respect to the camera frame

Camera & Model Transformations

- A “local frame view” is usually adopted as it extends naturally to the specification of hierarchical model frames.
- This allows creation of jointed assemblies
- articulated figures (animals, robots etc.)
- In the hierarchical model, each sub-component has its own local frame.
- Changes made to the parent frame are propagated down to the child frames (thus all models in a branch are globally controlled by the parent).
- This simplifies the specification of animation.

Aside: Display Lists

- It is often expensive to compute or create a model for display.
- we would prefer not to have to recalculate it each time for each new animation frame.
- Display lists allow the creation of an object in memory, where it resides until destroyed.
- Objects are drawn by issuing a request to the server to display a given display list.

number of list IDs to create

list = glGenLists(1);

glNewList(list, GL_COMPILE);

create_model();

glEndList();

glCallList(list);

glDeleteLists(list, 1);

begin specification of the list

GLint of first

list ID

user specified model code

used later to display the list

delete when finished

(specifying number of lists)

Model Transformations

- As the MODELVIEW matrix is changed objects are created with respect to a changing transformation.
- This is often termed the current transformation matrix or CTM.
- The CTM behaves like a 3D pointer, selecting new positions and orientations for the creation of geometries.
- The only complication with this view is that each new CTM is derived from a previous CTM, i.e. all CTMs are specified relative to previous versions.
- A scaling transformation can cause some confusion.

Scaling Transformation and the CTM

glMatrixMode(GL_MODELVIEW);

glLoadIdentity();

glTranslatef(3, 0, 0);

glTranslatef(1, 0, 0);

gluCylinder(…);

glMatrixMode(GL_MODELVIEW);

glLoadIdentity();

glTranslatef(3, 0, 0);

glScalef(0.5, 0.5, 0.5);

glTranslatef(1, 0, 0);

gluCylinder(…);

Hierarchical Transformations

- For geometries with an implicit hierarchy we wish to associate local frames with sub-objects in the assembly.
- Parent-child frames are related via a transformation.
- Transformation linkage is described by a tree:
- Each node has its own local co-ordinate system.

Hierarchical Transformations

R

R

R

T

Hierarchical transformation allow independent control over sub-parts of an assembly

OpenGL® Implementation

glMatrixMode(GL_MODELVIEW);

glLoadIdentity();

glTranslatef(bx, by, bz);

create_base();

glTranslatef(0, j1y, 0);

glRotatef(joint1_orientation);

create_joint1();

glTranslatef(0, uay, 0);

create_upperarm();

glTranslatef(0, j2y);

glRotatef(joint2_orientation);

create_joint2();

glTranslatef(0, lay, 0);

create_lowerarm();

glTranslatef(0, py, 0);

glRotatef(pointer_orientation);

create_pointer();

Hierarchical Transformations

- The previous example had simple one-to-one parent-child linkages.
- In general there may be many child frames derived from a single parent frame.
- we need some mechanism to remember the parent frame and return to it when creating new children.
- OpenGL provide a matrix stack for just this purpose:
- glPushMatrix() saves the CTM
- glPopMatrix() returns to the last saved CTM

Hierarchical Transformations

Each finger is a child of the parent (wrist)

independent control over the orientation of the fingers relative to the wrist

glLoadIdentity();

glTranslatef(bx, by, bz);

create_base();

glTranslatef(0, jy, 0);

glRotatef(joint1_orientation);

create_joint1();

glTranslatef(0, ay, 0);

create_upperarm();

glTranslatef(0, wy);

glRotatef(wrist_orientation);

create_wrist();

glPushMatrix(); // save frame

glTranslatef(-xf, fy0, 0);

glRotatef(lowerfinger1_orientation);

glTranslatef(0, fy1, 0);

create_lowerfinger1();

glTranslatef(0, fy2, 0);

glRotatef(upperfinger1_orientation);

create_fingerjoint1();

glTranslatef(0, fy3, 0);

create_upperfinger1();

glPopMatrix(); // restore frame

glPushMatrix();

// do finger 2...

glPopMatrix();

glPushMatrix();

// do finger 3...

glPopMatrix();

Finger1

OpenGL® Objects: GLU

- GLU provides functionality for the creation of quadric surfaces
- spheres, cones, cylinders, disks
- A quadric surface is defined by the following implicit equation:

use to initialise a quadric

GLUquadricObj* gluNewQuadric(void);

gluDeleteQuadric(GLYquadricObj *obj);

delete when finished

OpenGL® Objects: GLU Spheres

void gluSphere(GLUquadricObj *obj, double radius, int slices, int stacks);

gluSphere(obj, 1.0, 5, 5);

gluSphere(obj, 1.0, 10, 10);

gluSphere(obj, 1.0, 20, 20);

Other GLU Quadrics

void gluCylinder(GLUquadricObj *obj, double base_radius, double top_radius,

double height, int slices, int stacks);

gluCylinder(obj, 1.0, 1.0, 2.0, 20, 8);

gluCylinder(obj, 1.0, 1.0, 2.0, 8, 8);

gluCylinder(obj, 1.0, 0.3, 2.0, 20, 8);

Other GLU Quadrics

void gluDisk(GLUquadricObj *obj, double inner_radius, double outer_radius,

int slices, int rings);

gluCylinder(obj, 1.0, 0.0, 2.0, 20, 8);

gluDisk(obj, 0.0, 2.0, 10, 3);

gluDisk(obj, 0.5, 2.0, 10, 3);

OpenGL® Objects: GLUT

void glutSolidTorus(double inner_radius, double outer_radius,

int nsides, int rings);

glutWireTeapot(1.0)

glutWireTorus(0.3, 1.5,

20, 20);

size

glutSolidTorus(0.3, 1.5,

20, 20);

glutSolidTeapot(1.0)

glutSolidDodecahedron()

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