Assignment 7: Tangent Circles

**July 18, 2003**

I leave the bulk of the write-up on this geometry to GSP – it is an excellent tool for integrating the description with the working geometry and the number of diagrams for this one is excessive for the cut & paste routine.

A number of tabs are included in this **GSP file **to run. Generally a tab maps to one numbered problem
below, though some required multiple tabs and some none. In any case, the GSP page includes reference
to the problem number at the top.

GSP Scripts tools are saved in this file. They are kept separate to better copy into the \tools directory.

My angle on this (pun intended), is to begin notes on the 'so what' for the explorations. I hope to identify practical applications and interesting teaching points for each of these problems and solutions as I go through, though I suspect this quick pass may be lacking. If this section is short, it is because of the extent of time spent on the core constructions, but the intent is good and will leave work for another day.

I follow the outline structure of the problem sheet for easy mapping back.

As with all of the examples below, this provides opportunity to:

· Exercise geometrical intuition and understanding. The constructions reinforce core principles of circles, chords, triangles, bi-sectors, etc.

· Reinforce understanding of fundamental elliptical and hyperboloid shapes.

**Applications:**

· In general, these geometrical understandings serve as the basis for all sorts of mechanical designs, from gears and transmissions (some designed to generate elliptical or parabolic/hyperbolic movement from a rotating shaft. The designs of the 1780's to 1920's are often more innovative because they didn't have the option for electronic actuation and control.

· Engineers still make one of the highest starting salaries (that might get a few in the class).

· I believe elliptical and hyperboloid gear sets exercise some of these principles in a form.

· The rotary engine has a similar movement (I think) and uses the expansion and contraction of the 'constructed circle' for the purposes of a combustion chamber. Need to look into this.

· I believe this is the construction that served as the basis for the elliptical picture mat cutters.

This construction is very similar to – actually simpler than – the prior. Similar lessons and applications.

Construct the tangent circle to two given circles if the

- a. so that the smaller circle is external to the tangent circle.
- b. so that the smaller circle is internal to the tangent circle.

(*You may want to try the constructions from scratch
rather than using the script tools*.)

Don't yet get it.

In all cases in which the smaller circle remains within the larger, the center traces an ellipse.

Smaller circle is external to the tangent circle

If smaller circle is tangent to larger, then so is the ellipse as the constructed circle radius goes to 0 at point of tangency.

By definition, the ellipse is always outside of the smaller circle.

Smaller circle is internal to the tangent circle

Interestingly, in this case the ellipse can be inside, tangent, crossing, or outside of the inner given circle.

With center of inner given circle at the center of outer given circle, we construct a circle, of course

Interesting behavior with inner circle tangent to the outer circle and the outer circle having a diameter = radius of the outer circle. The elliptical pattern flattens into a line with focal points of the two centers of the given circles

5. & 6 & 7. Discuss the constructions of tangent circles if the two given circles intersect.

Smaller circle is external to the tangent circle

The ellipse always crosses the boundary of the two circles, by definition.

As inner circle crosses toward being completely outside the 'outer' circle (i.e. at tangency) the ellipse exhibits similar behavior as above for the internal circle. The ellipse flattens into a line and focal points move to the centers of the two given circles.

Smaller circle is internal to the tangent circle

As was suggested by it's 'trail-blazing' for the flattened ellipse, this construction leads the way in hyperbolic loci tracing. As soon as the inner circle crosses the boundary, a very 'narrow' hyperbola is traces with focal points near the two centers of the given circles. The hyperbola widens, and the focal points move away from the circle centers, as the inner circle travels further out across the bounds of the 'outer' circle.

If the two given circles are of equal size and they 'bisect' one another by having their circles pass through the center of the other, then the trace is a line between the two passing through their intersection point.

If the two given circle are of equal size and they are tangent to one another, then the trace is a line through the point of tangency. This aspect leads to the conjecture that the trace of any two equal size circles will be a line, which proves to be true.

It appears that the overall shape of the hyperbola is primarily driven by the relative size of the circles.

To prove that the locus of the centers, in each case, is the hyperbola with foci at the centers of the given circles.

Review graph to right. |HB| = |HG| by construction |BR| = |GT| by construction |BH| - |HC| = |CG| = |CT| + |TG| |CT| = radius of given circle C |TG| = radius of given circle B Since these are constants then the trace is a path of a parabola by geometric construction. |

9. Examine the trace of the tangent line in each case.

10. Consider the locus of the midpoint of the segment that formed
the base of the key isosceles triangle in each construction.

11. Using animation, consider the limit of the locus of centers as the two
given circles approach being tangent.

12. Is the locus ever a parabola? Is it a parabola in some limiting case?

13. Consider loci of points other than the center constructed relative to the
tangent circles. For example, consider points along the line through the point
of tangency to one of the given circles and the center. Or consider some points
along perpendiculars to the diameter through the point of tangency (try
perpendiculars not through the center as well as the perpendicular through the
center).

14. Given a line and a circle with center K. Take an arbitrary point P on the
circle. Construct two circles tangent to the given circle at P and tangent to
the line.

Investigate. . . what else can you find?

**Write-Up # 7.**

Prepare a retrospective summary on your experience with this assignment. The summary might take a mathematical bent, stressing the underlying theorems and relationships. It might take a pedagogical bent, stressing the exploration and discovery. It might take a "here is something interesting I found" bent. Or . . . be creative . . .