5.1: Surfaces and Their Representation

By 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

X(u,v) =x(u,v)E₁ +y(u,v)E₂ +z(u,v)E₃

The functions x(u,v) , y(u,v) and z(u,v) are called the coordinate functions of the surface.

Demonstration 1: Inputting Surfaces

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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

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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.