Triangulation

Introduction to Computer Graphics and Animation

(Principle of Computer Graphics)

Rattapoom Waranusast

Triangulations

• A valid triangulation of a figure X is a collection of triangles (T) satisfying the following two properties:

1. The collection T together exactly covers X.

2. Any given two triangles t1 and t2 in the collection are either

(a) disjoint (ie., do not intersect at all), or

(b) intersect exactly in a vertex of both, or

(c) intersect exactly in an edge (side) of both.

Examples

• Triangulation is not a unique process and the same object may have multiple triangulations.

Exercise 5.1

• Draw a square identical to the one in square.cpp, but using triangulation.– Use TRIANGLE_STRIP– Use TRIANGLE_FAN

• Draw a square and a triangle using triangulation with Steiner vertices.

• Draw a circle using triangulation– Without a Steiner vertex– With a Steiner vertex

Why should we care?

• If the collection satisfies the properties of a triangulation, then the image is independent of the order in which the triangles are rendered.

Experiment 5.1

• Run invalidTriangulation.cpp, press space bar to interchange the order of drawn vertices.

• Experiment with glutKeyboardFunc(keyInput) and keyInput(unsigned char key, int x, int y).

Quality of Triangulation

• Good triangulation– Few slivers– Most triangles being of nearly equal size– And each relatively small with respect to the entire

object

Quality of Triangulation

C

B

A

D

P

Triangulation by OpenGLand the Trouble with Non-Convexity

• Do the Experiment 4.16 again.

• When OpenGL is asked to draw a filled polygon, it creates a fan around the first vertex.

• Make sure to use GL_QUADS, GL_QUAD_STRIP, and GL_POLYGON with convex shape.

Experiment 5.2

• Replace the polygon declaration part of square.cpp with

glBegin(GL_POLYGON);glColor3f(1.0,1.0,1.0);

glVertex3f(80.0, 20.0, 0.0); glColor3f(1.0,0.0,0.0);

glVertex3f(40.0, 40.0, 0.0); glColor3f(1.0,1.0,1.0);

glVertex3f(20.0, 80.0, 0.0); glVertex3f(20.0, 20.0, 0.0); glEnd();

• The rendered ‘triangle’ is all white with no sign of red at all. Why?

Experiment 5.2

• Replace the polygon declaration part of square.cpp with

glBegin(GL_POLYGON);

glVertex3f(50.0, 10.0, 0.0);

glVertex3f(40.0, 50.0, 0.0);

glVertex3f(10.0, 60.0, 0.0);

glVertex3f(90.0, 60.0, 0.0);

glVertex3f(60.0, 50.0, 0.0);

glEnd();

• Predict the filled output each time as you rotate the vertices cyclically.

Experiment 5.2

• Replace the polygon declaration part of square.cpp with glBegin(GL_POLYGON);

glColor3f(1.0,1.0,1.0);glVertex3f(20.0, 20.0, 0.0);glColor3f(1.0,1.0,1.0);

glVertex3f(50.0, 20.0, 0.0);glColor3f(1.0,1.0,1.0);

glVertex3f(80.0, 50.0, 0.0); glColor3f(1.0,1.0,1.0);

glVertex3f(80.0, 80.0, 0.0);glColor3f(1.0,0.0,0.0); glVertex3f(20.0, 80.0, 0.0);

glEnd();

• Rotate the vertices by putting the first vertex (with its color) to the end of the list.

• Explain the difference in the rendering between the two listings?

Recommendation

• Avoid using GL_QUADS, GL_QUAD_STRIP, and GL_POLYGON.

• Use only the points, lines (including strip or loop) and triangles (including strip and fan) primitives– Convexity of individual piece is guaranteed– You control the quality of triangulation rather than

trusting the OpenGL’s fan-maker.

Vertex Array

• glVertex3fv(*pointer)• glColor3fv(*pointer)

• Separate data and code– Reduce redundancy and errors– Improve memory usage efficiency– Easy to modularize and debug

Experiment 5.3

• Run squareAnnulus1.cpp. Press the space bar to see the wireframe.

• Run squareAnnulus2.cpp. Compare the code with the previous experiment.

• Run squareAnnulus3.cpp. Compare the code with the previous experiments.

Initializing Vertex Array

• Vertex arrays are enabled with calls to glEnableClientState(array)

• The data for the arrays are specified with calls to glVertexPointer(size, type, stride, *pointer) and glColorPointer(size, type, stride, *pointer)

Experiment 5.3

• Run squareAnnulus4.cpp. Compare the code with the previous experiments.

• glDrawElements(primitive, count, type, *indices)

• glBegin(primitive)

for (i=0 ; i < count ; i++)

glArrayElement(indices[i]);

glEnd;

Assignment 2

1. Draw a face as in the picture. It need not be an exact replica but should be similar.

– Head and ears are circular– The user should be able to

toggle between the filled rendering and a wireframe via keyboard interaction.

– You can paint your face in any colors. A rainbow, if you like.

Assignment 2

2. Triangulate the figure "8". The "8" is thick. Color the figure as creatively as you like.

Assignment 2

• Note on assignment 2– Make sure you have your shapes valid triangulation– Allow the user to toggle between the filled rendering

and wireframe by pressing the space bar.– You are free to color objects as you like.– Instructions for user interaction should always be

output to the main C++ Window and also be written as comments at the top of your program file.