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OpenGL: Simple Use

- Open a window and attach OpenGL to it
- Set projection parameters (e.g., field of view)
- Setup lighting, if any
- Main rendering loop
- Set camera pose with gluLookAt()
- Camera position specified in world coordinates
- Render polygons of model
- Simplest case: vertices of polygons in world coordinates

Creating Geometry

- All geometry composed of vertices (points)
- ‘Primitive type’ defines the shape they describe
- Example vertices: glVertex (x, y, z, w)
- glVertex2d - z coordinate is set to 0.0
- glVertex3d - w coordinate is set to 1.0
- glVertex4d - all coordinates specified (rarely done)

Primitive Types

- GL_POINTS
- GL_LINE
- {S | _STRIP | _LOOP}
- GL_TRIANGLE
- {S | _STRIP | _FAN}
- GL_QUAD
- {S | _STRIP}
- GL_POLYGON

GL_POLYGON

- List of vertices defines polygon edges
- Polygon must be convex

Non-planar Polygons

- Imagine polygon with non-planar vertices
- Some perspectives will be rendered as concave polygons
- These concave polygons may not rasterize correctly

Generating Primitives

- Primitive defined within glBegin() and glEnd()
- Very few GL commands can be executed within these two GL calls
- Any amount of computation can be performed

glBegin (GL_LINE_LOOP);

for (j=0; j<10; j++) {

angle = 2*M_PI*j/10;

glVertex2f (cos(angle), sin(angle));

}

glEnd();

OpenGL: More Examples

- Example: GL supports quadrilaterals:

glBegin(GL_QUADS);

glVertex3f(-1, 1, 0);

glVertex3f(-1, -1, 0);

glVertex3f(1, -1, 0);

glVertex3f(1, 1, 0);

glEnd();

- This type of operation is called immediate-mode rendering; each command happens immediately

OpenGL: Front/Back Rendering

- Each polygon has two sides, front and back
- OpenGL can render the two differently
- The ordering of vertices in the list determines which is the front side:
- When looking at thefront side, the vertices go counterclockwise
- This is basically the right-hand rule

OpenGL: Drawing Triangles

- You can draw multiple triangles between glBegin(GL_TRIANGLES) and glEnd():

float v1[3], v2[3], v3[3], v4[3];

...

glBegin(GL_TRIANGLES);

glVertex3fv(v1); glVertex3fv(v2); glVertex3fv(v3);

glVertex3fv(v1); glVertex3fv(v3); glVertex3fv(v4);

glEnd();

- Each set of 3 vertices forms a triangle
- What do the triangles drawn above look like?
- How much redundant computation is happening?

OpenGL: Triangle Strips

- An OpenGL triangle strip primitive reduces this redundancy by sharing vertices:

glBegin(GL_TRIANGLE_STRIP);

glVertex3fv(v0);

glVertex3fv(v1);

glVertex3fv(v2);

glVertex3fv(v3);

glVertex3fv(v4);

glVertex3fv(v5);

glEnd();

- triangle 0 is v0, v1, v2
- triangle 1 is v2, v1, v3 (why not v1, v2, v3?)
- triangle 2 is v2, v3, v4
- triangle 3 is v4, v3, v5 (again, not v3, v4, v5)

v2

v4

v0

v5

v1

v3

Polygon Rendering Options

- Rendered as points, lines, or filled
- Front and back faces can be rendered separately using glPolygonMode( )
- glPolygonStipple( ) overlays a MacPaint-style overlay on the polygon
- glEdgeFlag specifies polygon edges that can be drawn in line mode
- Normal vectors: normalized is better, but glEnable(GL_NORMALIZE) will guarantee it

Polygonalization Hints

- Keep orientations (windings) consistent
- Best to use triangles (guaranteed planar)
- Keep polygon number to minimum
- Put more polygons on silhouettes
- Avoid T-intersections to avoid cracks
- Use exact coordinates for closing loops

E

BAD

OK

B

B

D

A

A

C

C

OpenGL: Specifying Normals

- Calling glNormal() sets the normal vector for the following vertices, till next glNormal()
- So flat-shaded lighting requires:

glNormal3f(Nx, Ny, Nz);

glVertex3fv(v0);glVertex3fv(v1);glVertex3fv(v2);

- While smooth shading requires:

glNormal3f(N0x, N0y, N0z); glVertex3fv(v0);

glNormal3f(N1x, N1y, N1z); glVertex3fv(v1);

glNormal3f(N2x, N2y, N2z); glVertex3fv(v2);

- (Of course, lighting requires additional setup…)

OpenGL: Specifying Color

- Calling glColor() sets the color for vertices following, until the next call to glColor()
- To produce a single aqua-colored triangle:

glColor3f(0.1, 0.5, 1.0);

glVertex3fv(v0); glVertex3fv(v1); glVertex3fv(v2);

- To produce a Gouraud-shaded triangle:

glColor3f(1, 0, 0); glVertex3fv(v0);

glColor3f(0, 1, 0); glVertex3fv(v1);

glColor3f(0, 0, 1); glVertex3fv(v2);

- In OpenGL, colors can also have a fourth component (opacity)
- Generally want = 1.0 (opaque);

OpenGL: Specifying Viewpoint

glMatrixMode(GL_MODELVIEW);

glLoadIdentity();

gluLookAt(eyeX, eyeY, eyeZ,

lookX, lookY, lookZ, upX, upY, upZ);

eye[XYZ]: camera position in world coordinates

look[XYZ]: a point centered in camera’s view

up[XYZ]: a vector defining the camera’s vertical

- Creates a matrix that transforms points in world coordinates to camera coordinates
- Camera at origin
- Looking down -Z axis
- Up vector aligned with Y axis

Translations

- For convenience we usually describe objects in relation to their own coordinate system
- We can translate or move points to a new position by adding offsets to their coordinates:
- Note that this translates all points uniformly

Scaling

- Scaling a coordinate means multiplying each of its components by a scalar
- Uniform scaling means this scalar is the same for all components:

2

Scaling

- Non-uniform scaling: different scalars per component:
- How can we represent this in matrix form?

2-D Rotation

x = r cos (f)

y = r sin (f)

x’ = r cos (f + )

y’ = r sin (f + )

Trig Identity…

x’ = r cos(f) cos() – r sin(f) sin()

y’ = r sin(f) sin() – r cos(f) cos()

Substitute…

x’ = x cos() - y sin()

y’ = x sin() + y cos()

(x’, y’)

(x, y)

f

2-D Rotation

- This is easy to capture in matrix form:
- 3-D is more complicated
- Need to specify an axis of rotation
- Simple cases: rotation about X, Y, Z axes

3-D Rotation

- What does the 3-D rotation matrix look like for a rotation about the Z-axis?
- Build it coordinate-by-coordinate

3-D Rotation

- What does the 3-D rotation matrix look like for a rotation about the Y-axis?
- Build it coordinate-by-coordinate

3-D Rotation

- What does the 3-D rotation matrix look like for a rotation about the X-axis?
- Build it coordinate-by-coordinate

3-D Rotation

- General rotations in 3-D require rotating about an arbitrary axis of rotation
- Deriving the rotation matrix for such a rotation directly is a good exercise in linear algebra
- Standard approach: express general rotation as composition of canonical rotations
- Rotations about X, Y, Z

Composing Canonical Rotations

- Goal: rotate about arbitrary vector A by
- Idea: we know how to rotate about X,Y,Z
- So, rotate about Y by until A lies in the YZ plane
- Then rotate about X by until A coincides with +Z
- Then rotate about Z by
- Then reverse the rotation about X (by -)
- Then reverse the rotation about Y (by -)

Composing Canonical Rotations

- First: rotating about Y by until A lies in YZ
- How exactly do we calculate ?
- Project A onto XZ plane (Throw away y-coordinate)
- Find angle to X:

= -(90° - ) = - 90 °

- Second: rotating about X by until A lies on Z
- How do we calculate ?

Composing Canonical Rotations

- Why are we slogging through all this tedium?
- A: Because you’ll have to do it on the test

3-D Rotation Matrices

- So an arbitrary rotation about A composites several canonical rotations together
- We can express each rotation as a matrix
- Compositing transforms == multiplying matrices
- Thus we can express the final rotation as the product of canonical rotation matrices
- Thus we can express the final rotation with a single matrix!

Compositing Matrices

- So we have the following matrices:

p: The point to be rotated about A by

Ry : Rotate about Y by

Rx : Rotate about X by

Rz : Rotate about Z by

Rx -1: Undo rotation about X by

Ry-1: Undo rotation about Y by

- In what order should we multiply them?

Compositing Matrices

- Short answer: the transformations, in order, are written from right to left
- In other words, the first matrix to affect the vector goes next to the vector, the second next to the first, etc.
- So in our case:

p’ = Ry-1 Rx -1 Rz Rx Ry p

Rotation Matrices

- Notice these two matrices:

Rx : Rotate about X by

Rx -1: Undo rotation about X by

- How can we calculate Rx -1?

Rotation Matrices

- Notice these two matrices:

Rx : Rotate about X by

Rx -1: Undo rotation about X by

- How can we calculate Rx -1?
- Obvious answer: calculate Rx (-)
- Clever answer: exploit fact that rotation matrices are orthonormal

Rotation Matrices

- Notice these two matrices:

Rx : Rotate about X by

Rx -1: Undo rotation about X by

- How can we calculate Rx -1?
- Obvious answer: calculate Rx (-)
- Clever answer: exploit fact that rotation matrices are orthonormal
- What is an orthonormal matrix?
- What property are we talking about?

Rotation Matrices

- Orthonormal matrix:
- orthogonal (columns/rows linearly independent)
- normalized (columns/rows length of 1)
- The inverse of an orthogonal matrix is just its transpose:

Modeling Transformations

- glTranslate (x, y, z)
- Multiplies the current matrix by a matrix that moves the object by the given x-, y-, and z-values
- glRotate (theta, x, y, z)
- Multiplies the current matrix by a matrix that rotates the object in a counterclockwise direction about the ray from the origin through the point (x, y, z)

Modeling Transformations

- glScale (x, y, z)
- Multiplies the current matrix by a matrix that stretches, shrinks, or reflects an object along the axes.

Matrix Multiplcations

- Certain commands affect the current matrix in OpenGL
- glMatrixMode() sets the current matrix
- glLoadIdentity() replaces the current matrix with an identity matrix
- glTranslate()postmultipliesthe current matrix with a translation matrix
- gluPerspective() postmultipliesthe current matrix with a perspective projection matrix
- It is important that you understand the order in which OpenGL concatenates matrices

Matrix Operations In OpenGL

- In OpenGL:
- Vertices are multiplied by the MODELVIEW matrix
- The resulting vertices are multiplied by the projection matrix
- Example:
- Suppose you want to scale an object, translate it, apply a lookat transformation, and view it under perspective projection. What order should you make calls?

Matrix Operations in OpenGL

- Problem: scale an object, translate it, apply a lookat transformation, and view it under perspective
- A correct code fragment:

glMatrixMode(GL_PERSPECTIVE);

glLoadIdentity();

gluPerspective(…);

glMatrixMode(GL_MODELVIEW);

glLoadIdentity();

gluLookAt(…);

glTranslate(…);

glScale(…);

/* Draw the object... */

Matrix Operations in OpenGL

- Problem: scale an object, translate it, apply a lookat transformation, and view it under perspective
- An incorrect code fragment:

glMatrixMode(GL_PERSPECTIVE);

glLoadIdentity();

glTranslate(…);

glScale(…);

gluPerspective(…);

glMatrixMode(GL_MODELVIEW);

glLoadIdentity();

gluLookAt(…);

/* Draw the object... */

Multiplication Order

glMatrixMode (MODELVIEW);

glLoadIdentity();

glMultMatrix(N);

glMultMatrix(M);

glMultMatrix(L);

glBegin(POINTS);

glVertex3f(v);

glEnd();

Modelview matrix successively contains:

I(dentity), N, NM, NML

The transformed vertex is:

NMLv = N(M(Lv))

Multiplication Order

- Rotate line segment by 45 degrees about endpoint

Wrong

Right

R(45)

T(-3), R(45), T(3)

OpenGL

T(3)

R(45)

T(-3)

Manipulating Matrix Stacks

- Observation: Certain model transformations are shared among many models
- We want to avoid continuously reloading the same sequence of transformations
- glPushMatrix ( )
- push all matrices in current stack down one level and copy topmost matrix of stack
- glPopMatrix ( )
- pop the top matrix off the stack

Matrix Manipulation - Example

- Drawing a car with wheels and lugnuts

draw_wheel( );

for (j=0; j<5; j++) {

glPushMatrix ();

glRotatef(72.0*j, 0.0, 0.0, 1.0);

glTranslatef (3.0, 0.0, 0.0);

draw_bolt ( );

glPopMatrix ( );

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