5.1: Surfaces and Their RepresentationBy a surface in three-space, we mean a mapping of a domain in the plane into ordinary three-dimensional Euclidean space. In most of our examples, the domain of a surface will be a rectangle, and can be considered with or without its boundary edges. | |
We will consider the points of three-dimensional space as vectors emanating from the origin. A surface is usually represented as a parametric vector function X of two parameters (u,v) . For a particular value (u0,v0) in the domain of the function, the vector corresponding to the point (u,v) in the domain is denoted by X(u,v). | |
In terms of the standard basis E1 ,E2 ,E3 of Euclidean three-space, we have The functions x(u,v) , y(u,v) and z(u,v) are called the coordinate functions of the surface. | |
Demonstration 1: Inputting Surfaces The default function is a function graph given by x(u,v)=u , y(u,v)=v and z(u,v)=u2-v2 , with domain given by the rectangle where u and v are between -1 and 1. The demonstration shows the domain in one window and the surface in three-dimensional space in the second window. Selecting a point in the domain highlights a small rectangle, and simultaneously shows the image of that rectangle on the surface. | |
As in the case of curves, we can enter in the equations of the coordinate functions of a surface, and the values defining the rectangular domain of these functions, and the program will display the surface in three-dimensional space. | |
If we fix a value of v=v0 , then as u changes, we have a curve X(u,v0) on the surface called a u-parameter curve . Similarly, if we fix a value of u=u0 , then as v changes, we have a curve X(u0,v) on the surface called a v-parameter curve . | |
Demonstration 2: Parameter Curves Demo To display the parameter curves, there are two checkboxes which allow you to toggle the display of either one. Below each checkbox is a tapedeck that controls the position of the u- or v-parameter curve by scaling between the minimum and the maximum of the domain in u or the domain in v . In the bottom right corner of the control panel, the actual coordinates of the point are displayed together. | |
As we move a point around in the domain, the demonstration shows how the u- and v-parameter curves through that point move around on the surface. |