Activities

Movie 1

The figure shows the point P0 and the ridge formed by the two intersecting planes. The shortest distance between P0 and the ridge would be the radius of the sphere that is centered at P0 and to which the ridge is a tangent. Push the play button on figure 1. The animation shows a spherical balloon that expands gradually, until it just touches the ridge.

Movie 1

Movie 2

The animation in figure 2 simply rotates the angle of view to a head-on perspective, so that the tangency is clearly displayed. The problem is to determine, analytically, the point of tangency P.
Movie 2

Movie 3

We now explore further the geometry of the tangency condition at the optimal point P.

Figure 3 shows three vectors originating at P; two of these are the normal vectors to the two planes, and the third is normal to the sphere.

Respond to the following on the worksheet.
  • How do you compute the three vectors?

Play the animation in figure 3 manually, by moving the slider gradually to the right. As the figure rotates, you are better able to observe how the three vectors are oriented relative to one another.

Respond to the following on the worksheet.
  • What do you discover about the relative orientation of the three vectors? Why is the relative orientation the way it is?
  • How do you express the mutual relationship among the three vectors algebraically?

Movie 3

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