Meshing and Geometry

Points in OpenGL

glBegin(GL_POINTS);

glVertex2fv(p0);

glVertex2fv(p1);

glVertex2fv(p2);

glVertex2fv(p3);

glVertex2fv(p4);

glVertex2fv(p5);

glVertex2fv(p6);

glVertex2fv(p7);

glEnd();

p0 p1

p2

p3

p4

p5

p6

p7

Lines in OpenGL (1/3)

Line Segments glBegin(GL_LINES);

glVertex2fv(p0);

glVertex2fv(p1);

glVertex2fv(p2);

glVertex2fv(p3);

glVertex2fv(p4);

glVertex2fv(p5);

glVertex2fv(p6);

glVertex2fv(p7);

glEnd();

p0 p1

p2

p3

p4

p5

p6

p7

Lines in OpenGL (2/3)

Polylines – Line Strip glBegin(GL_LINE_STRIP);

glVertex2fv(p0);

glVertex2fv(p1);

glVertex2fv(p2);

glVertex2fv(p3);

glVertex2fv(p4);

glVertex2fv(p5);

glVertex2fv(p6);

glVertex2fv(p7);

glEnd();

p0 p1

p2

p3

p4

p5

p6

p7

Lines in OpenGL (3/3)

Polylines – Line Loop glBegin(GL_LINE_LOOP);

glVertex2fv(p0);

glVertex2fv(p1);

glVertex2fv(p2);

glVertex2fv(p3);

glVertex2fv(p4);

glVertex2fv(p5);

glVertex2fv(p6);

glVertex2fv(p7);

glEnd();

p0 p1

p2

p3

p4

p5

p6

p7

Polygons (1/2)

Definition Object that is closed as a line loop, but that has an interior (Fill is guaranteed for convex polygon)

Simple Polygon No pair of edges of a polygon cross each other

Simple Nonsimple

Polygons (2/2)

Convexity If all points on the line segment between any two points inside the object, or on its boundary, are inside the object

Convex Objects

p1

p2

Polygons in OpenGL (1/6)

Polygon glBegin(GL_POLYGON);

glVertex2fv(p0);

glVertex2fv(p1);

glVertex2fv(p2);

glVertex2fv(p3);

glVertex2fv(p4);

glVertex2fv(p5);

glVertex2fv(p6);

glVertex2fv(p7);

glEnd();

p0 p1

p2

p3

p4

p5

p6

p7

Polygons in OpenGL (2/6)

Quadrilaterals glBegin(GL_QUADS);

glVertex2fv(p0);

glVertex2fv(p1);

glVertex2fv(p2);

glVertex2fv(p3);

glVertex2fv(p4);

glVertex2fv(p5);

glVertex2fv(p6);

glVertex2fv(p7);

glEnd();

p0 p1

p2

p3

p4

p5

p6

p7

Polygons in OpenGL (3/6)

Quadstrip glBegin(GL_QUAD_STRIP);

glVertex2fv(p1);

glVertex2fv(p2);

glVertex2fv(p3);

glVertex2fv(p0);

glVertex2fv(p4);

glVertex2fv(p7);

glVertex2fv(p5);

glVertex2fv(p6);

glEnd();

p0 p1

p2

p3

p4

p5

p6

p7

Polygons in OpenGL (4/6)

Triangles glBegin(GL_TRIANGLES);

glVertex2fv(p0);

glVertex2fv(p1);

glVertex2fv(p2);

glVertex2fv(p3);

glVertex2fv(p4);

glVertex2fv(p5);

glVertex2fv(p6);

glVertex2fv(p7);

glEnd();

p0 p1

p2

p3

p4

p5

p6

p7

Polygons in OpenGL (5/6)

Triangle Strip glBegin(GL_TRIANGLE_STRIP);

glVertex2fv(p0);

glVertex2fv(p7);

glVertex2fv(p1);

glVertex2fv(p6);

glVertex2fv(p2);

glVertex2fv(p5);

glVertex2fv(p3);

glVertex2fv(p4);

glEnd();

p0 p1

p2

p3

p4

p5

p6

p7

Polygons in OpenGL (6/6)

Triangle Fan glBegin(GL_TRIANGLE_FAN);

glVertex2fv(p0);

glVertex2fv(p1);

glVertex2fv(p2);

glVertex2fv(p3);

glVertex2fv(p4);

glVertex2fv(p5);

glVertex2fv(p6);

glVertex2fv(p7);

glEnd();

p0 p1

p2

p3

p4

p5

p6

p7

triangle code

void triangle( point a, point b, point c)

{

glBegin(GL_POLYGON);

glVertex3fv(a);

glVertex3fv(b);

glVertex3fv(c);

glEnd();

}

subdivision code

void divide_triangle(point a, point b, point c, int

m)

{

point v1, v2, v3;

int j;

if(m>0)

{

for(j=0; j<3; j++) v1[j]=(a[j]+b[j])/2;

for(j=0; j<3; j++) v2[j]=(a[j]+c[j])/2;

for(j=0; j<3; j++) v3[j]=(b[j]+c[j])/2;

divide_triangle(a, v1, v2, m-1);

divide_triangle(c, v2, v3, m-1);

divide_triangle(b, v3, v1, m-1);

}

else(triangle(a,b,c));

}

tetrahedron code

void tetrahedron( int m)

{

glColor3f(1.0,0.0,0.0);

divide_triangle(v[0], v[1], v[2], m);

glColor3f(0.0,1.0,0.0);

divide_triangle(v[3], v[2], v[1], m);

glColor3f(0.0,0.0,1.0);

divide_triangle(v[0], v[3], v[1], m);

glColor3f(0.0,0.0,0.0);

divide_triangle(v[0], v[2], v[3], m);

}

Almost Correct

Because the triangles are drawn in the order they are defined in the program, the front triangles are not always rendered in front of triangles behind them

get this

want this

Hidden-Surface Removal

We want to see only those surfaces in front of other surfaces

OpenGL uses a hidden-surface method called the z-buffer algorithm that saves depth information as objects are rendered so that only the front objects appear in the image

Using the z-buffer algorithm

The algorithm uses an extra buffer, the z-buffer, to store depth information as geometry travels down the pipeline

It must be Requested in main.c

-glutInitDisplayMode

(GLUT_SINGLE | GLUT_RGB | GLUT_DEPTH)

Enabled in init.c -glEnable(GL_DEPTH_TEST)

Cleared in the display callback -glClear(GL_COLOR_BUFFER_BIT |

GL_DEPTH_BUFFER_BIT)

Meshlab

1. Model loading (points, edges, meshes)

2. Orthographic

3. Shader

4. Filters (Smoothing)

5. Lighting (Ctrl+shift+left mouse)

Delaunay triangulation

1. For any triangle in Delaunay triangulation, the circle passing through its three vertices has no other vertices in its interior.

2. For any edge in the Delaunay triangulation, there is no circle passing through the endpoints (vertices) of this edge that includes another vertex in its interior.

3. If we consider the set of angles of all the triangles in a triangulation, the Delaunay triangulation has the greatest minimum angle.

Delaunay triangulation

Delaunay triangulation

Delaunay triangulation

- The act of flipping one edge can require the

flipping of other edges

- The triangulation has a O(n log n) complexity

https://www.youtube.com/watch?v=1uYI8T5dvf0

https://www.youtube.com/watch?v=_1M4t73L4EE

https://www.youtube.com/watch?v=xzmg0grXHyE