3d surfaces

8/2/2019 3d Surfaces

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3D SURFACES

Planar – A flat, two-dimensional bounded surface. A planar

surface can be defined as the motion of a straight-

line generatrix that is always in contact with either

two parallel straight lines, two intersecting lines, ora line and a point not on the line.

Single-curved - The simple-curved bounded face of an object

produced by a straight-line generatrix revolved

around an axis directrix (yielding a cylinder) or avertex directrix (yielding a cone).

Double-curved - Contains no straight lines and is the compound-

surved bounded face of an object produced by

an open or closed curved-line generatrix revolvedaround an axis directrix (yielding a sphere or

ellipsoid), a center directrix (yielding a torus), or a

vertex directrix (yielding a paraboloid or a

hyperboloid).

Warped – A single- and double-curved transistional surface

(cylindroid, conoid, helicoids, hyperbolic

paraboloid), often approximated by triangulated

surface sections that may join other surfaces or

entities together.



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Freeform – Follows no set pattern and requires

more sophisticated underlying

mathematics.

Ruled – Produced by the movement of a straight-

line generatrix controlled by a directrix to

form a planar, single-curved, or warped

surface

Developable – Can be unfolded or unrolled onto a plane

without distortion. Single-curved

surfaces, such as cylinders and cones,

are developable

Undevelopable – Cannot be unfolded or unrolled onto a

plane without distortion. Warped and

double-curved surfaces, such as a sphere

or an ellipsoid, cannot be developed except

by approximation. For example, the Earth isnearly a sphere, and methods that

represent its land forms on flat paper have taken cartographers

centuries to develop. On some types of maps, the land forms near

the poles of the Earth are drawn much larger than they really are, to

compensate for the curvature that can’t be shown.



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CLASSIFICATION

Coons’ – Constructed from a mesh of input curves. The

surface was named after Steven A. Coons, who

developed the mathematical method for defining

complex shapes, such as those used in the designof aricraft, automobiles, and ships’ hulls. The

simplest form of Coons’ surface is a single patch,

which is very limiting. Of more practical value is a

smooth surface made up of a composite blend of multiple patches.

The advantage of usong Coons’ surface is its ability to it a smooth

surface exactly throughout digitized or wireframe data. The weakness

is that, to change the shape of the surface, the input curves must be

changed. A single Coons’ surface patch is constructed by blending

four boundary curves.

Bezier – A surface that can be shaped by manipulation of the

control point, which can be pulled to alter the shape of the

surface. This means the surface is easy to modify,

making it ideal for desing changes. However, pulling a

single control point will alter the entire surface; therefore,

the editing of control points on Bezier surfaces has global

effects similar to that of a Bezier curve.



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B-Spline - Has the strong points of both the Coon’s

and Bezier surfaces. It can be mapped

to a mesh of existing data points, which

can be pulled to edit the shape of the

surface. However, pulling the control

points does not affect the border curves.

A major advantage of B-Spline surfaces

over Bezier surfaces is that, when a

control point is pulled, it only affects the

immediate area; that is, you have local

control over the surface shape, similar to the local effects on a B-

spline curve. These characteristics make the B-spline surface useful

in an interactive modeling environment.

NURBS - Nonuniform Rational B-spline, the most

advanced form of surface mathematics. As

the name implies, NURBS include all the

characteristics of B-spline surfaces, with

additional capabilities. The difference

between NURBS and B-spline surfaces is

that the NURBS is rational; that is, for every point described on the

surface, there is an associated weight. Dividing the surface pointfunction by the weight function creates new surfaces that are not

possible with nonrational surfaces. Note that if the weight factor is one

for all the points, the surface is a B-spline. The

greatest application for NURBS is for nonregular

free-form surfaces that cannot be defined using

traditional methods. Surfaces based on arcs or

conics can be represented exactly by a NURBS

surface. B-spline surfaces and others can only

approximate these shapes, using numerous

small patches.



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Fractals – Short for fractional dimensional, to define

such repetition mathematically. Fractal

geometry has led to the development of

computer-based fractal design tools that

can produce very complex random

patterns. The term fractal is used to

describe graphics of randomly generated

curves and surfaces that exhibit a degree of self-similarity. These

fractal curves and surfaces emerge in the form of images that are

much more visually realistic than can be reproduced with conventional

geometric forms. In addition, fractal image compression techniques

can be used to solve storage problems associated with large

graphical image files on computers.

3d surfaces

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