Computer Graphics. Chapter 7 3D Object Modeling. 3D Object Representation. y = f(x,z) . A surface can be analytically generated using its function involving the coordinates. An object can be represented in terms of its vertices, edges and polygons. (Wire Frame, Polygonal Mesh etc.)
3D Object Modeling
y = f(x,z)
x y z . . .
V + FE = 2.
y = f(x, z)
Many surfaces can be represented by an explicit function of two independent variables, such as y = f(x, z).
A polyhedron obtained by sweeping (extruding) a polygon along a straight line is called a prism.
y = f(x)
y = f(r)
The three-dimensional surface obtained by revolving the curve y = f(x) about the y-axis is obtained by replacing x with sqrt(x*x + z*z).
The surface of revolution is thus given by
Quad trees are generated by successively dividing a 2-D region(usually a square) into quadrants. Each node in the quadtree has 4 data elements, one for each of the quadrants in the region. If all the pixels within a quadrant have the same color (a homogeneous quadrant), the corresponding data element in the node stores that color. For a heterogeneous region of space, the successive divisions into quadrants continues until all quadrants are homogeneous.
The Bezier curve only approximates the control points
and doesn’t actually pass through all of them.
Inputs: n control points (xi, yi), i = 0, 1,2, …n-1
m = n1
Inputs: n control points (xi, yi), i = 0, 1,2, …m
p2Properties of Bezier Curve
p0Properties of Bezier Curve (cont)
p0=p5Design Technique using Bezier Curves:
p1 = p2
p4Design Technique (Cont)
A set of 16 control points
The Bezier Patch
Utah Teapot Defined Using Control Points
Utah Teapot Generated Using Bezier Patches