Geometry

Dynamical Geometry Problem Sheet 3: Circle and ellipse

Sketches: Martti E. Pesonen
Problems: ?
Let E be the two dimensional Euclidean plane. Let us assume points, lines and segments to be familiar, also the notion of distance assumed to be known.
NOTE! If you are not yet familiar with
1) Move and Animate facilities,
2) resetting the sketch and clearing traces,
you are advised to consult

Example Sketch
Getting familiar with Sketches: experimenting with circles

The forthcoming sketches contain various kinds of mathematical tasks.
Some of them require more explicit or formal mathematical reasoning, and an unprejudiced attitude for experimentation!

Sketch 1. Ellipse by definition, fixed focus points

Let F and G be a two points and AB a segment in the plane. An ellipse with focus points F and G and joint distance AB is the set of all points X, for which the distance from F to G via X is equal to the length of AB.

The following dynamical figure illustrates how the ellipsis is built from its definition. Here the focus points are fixed, but the segment representing the sum of distances |AB| can be changed by mouse.

This page requires a Java capable browser.
You can see the trace of the point that draws the ellipse, when you slide the red point that divides the total length to the pieces, focus rays.
At first, the geometrical construction is hidden, but can be shown with a button push.

Sketch 2: Ellipse by definition, with draggable focus points

One can very well imagine how the circle comes as a special case, by moving the focus points on top of each other. Try it.
Here we meet problems. Can you propose an explanation? This page requires a Java capable browser.
$LAB Line($B,$A)[red,hidden]; $AA Point(200,450)[red]; $BB VectorTranslation($AA,$A,$B)[red]; $SAABB Segment($AA,$BB)[red];