Lecture 03b: Shapes

COMP 175 | COMPUTER GRAPHICS

Remco Chang 1/6103b – Shapes

Lecture 03b:

Shapes

COMP 175: Computer GraphicsFebruary 9, 2016



Points vs. Vectors

What are basis vectors? In 2D, what 2 vectors cannot be

basis vectors?

What is: a unit vector a normal vector a null vector?

Vector math: Add, subtract, multiple (by scalar) Normalize a vector

What is a cross product? How do you find the cross

product? Different ways to think about the

cross product?

What is a dot product? How do you find the dot product? Different ways to think about the

dot product?

Given two vectors A, B, show that the cross product N is orthogonal to A and B.

Quick Review of Linear Algebra



Lines and Polyline Polyline: lines drawn between

ordered points

Polygon If the first and last points of a polyline

are the same point, then the polyline is said to be “closed”, and that the closed polyline forms a polygon

Self-intersected polyline

2D Object Definition

simple polygon,closed polyline

not closed, simple polyline

not simple polygon, closed polyline



Convex and Concave polygons Convex: for every pair of points inside the polygon, the

line between them is entirely inside the polygon

Concave: for some pair of points inside the polygon, the line between them is not entirely inside the polygon. Not convex.


convex

concave



Why is a convex polygon so awesome?




Why is a convex polygon so awesome? Center of mass Tessellation Subdivision Fixed boundary Exactly two intersections All diagonals are within the boundary Tessellation -> computing the area Every interior angle is less than 180 Etc.




Circle Set of all points equidistant from one point called the center The distance from the center is the radius r The equation for a circle centeredat (0, 0) is r2 = x2 + y2

Special Polygons

Triangle Square Rectangle

(0, y)

(0, x)

r

(0, 0)

(x, y)



A circle can be approximated by a polygon with many sides.

Axis aligned ellipse: a circle scaled in the x and/or y direction


0 1

1

2

2

3 4 5 6 7 8 9 10

3

4

5

6

0 1

1

2

2

3 4 5 6 7 8 9 10

3

4

5

6



Vertex and Edge tables General purpose, minimal overhead, reasonably efficient Each vertex listed once Each edge is an ordered pair of indices to the vertex list

Representing Shapes

Vertices

0 (0, 0)

1 (2, 0)

2 (0, 1)

3 (2, 1)

4 (1, 1.5)

Edges

0 (0, 1)

1 (1, 3)

2 (3, 4)

3 (4, 2)

4 (2, 0)

E4

E3 E2

E0

E1

V0 V1

V2 V3

V4



Most common representation of shape in three dimensions

All vertices of triangle are guaranteed to lie in one plane (not true for quadrilaterals or other

polygons) Uniformity makes it easy to

perform mesh operations such as subdivision, simplification, transformation etc.

Many different ways to represent triangular meshes

Triangle Meshes



Vertex, Face, (and Normal) Tables

Triangular Mesh Representation

Vertex List

v0 0, 0, 0 f0 f1 f12 f15 f7

v1 1, 0, 0 f2 f3 f13 f12 f1

v2 1, 1, 0 f4 f5 f14 f13 f3

v3 0, 1, 0 f6 f7 f15 f14 f5

v4 0, 0, 1 f6 f7 f0 f8 f11

v5 1, 0, 1 f0 f1 f2 f9 f8

v6 1, 1, 1 f2 f3 f4 f10 f9

v7 0, 1, 1 f4 f5 f6 f11 f10

v8 .5, .5, 0

f8 f9 f10 f11

v9 .5, .5, 1

f12 f13 f14 f15

Face List

f0v0 v4

v5

f1v0 v5

v1

f2v1 v5

v6

f3v1 v6

v2

f4v2 v6

v7

f5v2 v7

v3

f6v3 v7

v4

f7v3 v4

v0

f8v8 v5

v4

f9v8 v6

v5

f10v8 v7

v6

f11v8 v4

v7

f12v9 v5

v4

f13v9 v6

v5

f14v9 v7

v6

f15v9 v4

v7

v0

v1 v2

f3

f2

f1

f0

f9

v8

v4 v7

v6

f10f8f11

v5



Questions?



Normals



Calculating Normal



3D Barn Shape



3D Barn Shape



3D Barn Shape



3D meshes can be represented as vertices, (edges), faces, and normals

Are there other ways to encode 3D meshes?

Other Possible Encodings



Parametric Shapes



2.5D objects are 3D objects that are “protrusions” or “extensions” of 2D shapes (i.e., 2D shape + height)

2.5D Objects



3D Extrusions



A spring can be described in Cartesian coordinate system as:

Or in cylindrical coordinate as:

3D Extrusions



Surfaces from Revolutions



We can think of the shapes in assignment 2 the same way... ““Vertices in Motion”

So we can make 3D shapes out of 2D and 1D primitives

Technically...



Questions?



These two have the same number of quads… What is the difference?

GL Normal



GL Normal



glNormal3f(nx, ny, nz);glBegin(GL_QUADS); glVertex3f(x1, y1, z1); glVertex3f(x2, y2, z2); glVertex3f(x3, y3, z3); glVertex3f(x4, y4, z4);glEnd();

GL Normal



glBegin(GL_QUADS); glNormal3f(nx1, ny1, nz1); glVertex3f(x1, y1, z1);

glNormal3f(nx2, ny2, nz2); glVertex3f(x2, y2, z2);

glNormal3f(nx3, ny3, nz3); glVertex3f(x3, y3, z3);

glNormal3f(nx4, ny4, nz4); glVertex3f(x4, y4, z4);glEnd();

GL Normal



What are the normals? For a circle / sphere? For a cube? For a cylinder? For a cone?

How do you define what a normal is?

GL Normal



What are the normals? For a circle / sphere? For a cube? For a cylinder? For a cone?

Thinking of the normal of a curved surface as the normal to the tangent plane

GL Normal



Write a render function that takes in two parameters, radius and slices, and produces a 2D circle drawn as a set of polylines (assume centered at (0,0)), with a defined normal for each vertex

That is, fill in this function:

void drawCircle2D(float radius, int slices) { : : glBegin (...); for (int i=0; i<...) { : : glNormal2f(...); glVertex2f(...); } glEnd();}

Exercise



Questions?

Documents

Lecture 03b: Shapes