In my precalc class, we were introduced to conics with a very cool activity involving a square piece of wax paper and a protractor. The activity worked like so:

  1. Draw a circle roughly in the center of the wax paper with a radius of about a quarter of the paper length (just make sure the circle isn’t huge)
  2. Mark a point somewhere on the paper
  3. Continue to fold the paper so that the marked point lies on the circle until you are pleased with the resulting image

Depending on where you put the point (around the circle, far from the circle, on or the center of the circle), you’ll see a different conic (an ellipse, hyperbola, or circle respectively). If you want to draw a parabola just use a line instead of a circle. This generates what is called an envelope of a conic, which is just a set of lines which are each tangent to the conic. The reason this works for the ellipse is that an ellipse can be defined given two points (the foci) and a distance as the locus of all points for which the sum of the distances from each foci is constant. In the case of the wax paper the two foci are the chosen point and the center of the circle and the distance is the radius of the circle. When you fold the marked point to the circle, the distance from the center of the circle to the fold line plus the distance from the fold line to the point is always equal to the radius of the circle. A similar idea explains why choosing a different point will produce different conics.

I thought this conic envelope thing was pretty cool so I decided to write a little visualization for it using p5.js to render the graphics. I was actually pretty surprised with how simple the code for this was. All of the actual work takes place in this little loop:

for(var rads = 0; rads < TWO_PI; rads+=epsilon){
    //Generate a point on a circle with a radius of one fourth of the smallest
    //screen dimension and a center at the center of the screen
    var x = SMALL/4*Math.cos(rads) + WIDTH/2;
    var y = SMALL/4*Math.sin(rads) + HEIGHT/2;

    //p is the midpoint between the marked focus and the current point on the
    //circle (x, y)
    var p = {x:(x+focus.x)/2, y:(y+focus.y)/2};
    //This is the slope of a line perpindicular to the line between the focus
    //and (x, y)
    var slope = -1*(focus.x - x)/(focus.y - y);
    //Draw a line through point p with slope slope (recognize the point slope
    var farLeftY = slope*(0 - p.x) + p.y; 
    var farRightY = slope*(WIDTH - p.x) + p.y;
    line(0, farLeftY, WIDTH, farRightY);

This algorithm works by moving a point around a circle and then drawing the perpendicular bisector of that point and the given point. This geogebra visualization I made shows how and why that works.


Explain possible improvements

Explain in more depth why algo works