5.5: Areas of Regions

To find the area of a domain on a surface, we compute the double integral of the areas of the parallelograms with sides given by the partial derivative vectors. Thus the area of a region is the integral of the length of the cross product, i.e. the integral of the square root of g₁₁(u,v)g₂₂(u,v)-g₁₂( u,v)²=g(u,v) . This function represents how much the domain is being stretched at a given point.

Demonstration 6: Area Function Graph Demo

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In this demo, The Area Element Graph shows that area element graphed over the domain. This means that we draw the function graph for g(u,v) . The total area is shown in the readout in the bottom of the control window.

Demonstration 6a: Surface Colored By Area

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Here's a similar demo, except that instead of graphing the area element separately, the area element at each point is used to select the color for that point. You choose the maximum value of the area used for coloring (the value that's represented as color 1). Any areas greater than this value receive colors at the beginning of the spectrum again.

You may notice that the value of the area is slightly overestimated for compact regions, because fnord counts bits of the edges of the domain twice where they come together on the surface. You can minimize this problem by increasing the number of divisions in the domain.