Cynthia Schneider
Math 588
Assignment Seven
Problem One: Parabola
The construction of the parabola using the standard formula 
where (h,k) = the
vertex coordinate, and p = the distance from focus to vertex. Then the directrix
is y = k – p.
For my first parabola I used the following values: h = 0, k
= 0, p = 2, and y = -2 for the directrix. This yields the equation 
or y = 
.
Using Geometer’s Sketchpad we can graph the equation.
Now we construct the locus of points for the focus (0,2) and the directrix y = -2 with Geometer’s Sketchpad.
Create a moving but dependent point p on the directrix.
Create a line through the point and perpendicular to the directrix.
Construct a line segment from p to the focus.
Construct the midpoint of this line segment.
Construct a line through the point and perpendicular to the line segment.
Construct the intersection, point t, of this line and the line though p.
We will trace this intersection point t.
We animate the point p along the directrix.
The results are a locus of points traced out over the top of our constructed parabola
y =.
Here is a link to the Geometer’s Sketchpad construction file.
Problem Two: Pedal
Curves
The following pedal curves are for the parabola function
, with a focus at the point (-4, 2), a vertex at the point
(-4, 0), and a directrix y = -2.
Here is a pedal curve with the pedal point at the focus.
The pedal curve is a straight line.
Here is a pedal curve with the pedal point at the vertex.
Since the graph does not continue past x = -12, the pedal curve is not symmetric. If we changed the graph to include smaller negative values we would see that the pedal curve is symmetric.
Here is a pedal curve with the pedal point on the directrix.
If we change the position of the pedal point we obtain different pedal curves such as this.
Here is another pedal curve with the pedal point on the directrix.
Finally we have a pedal curve with the pedal point on the reflection point of the focus by the directrix.
Here again if our negative x values continued on we would see a symmetric curve about the line x = -4.
The procedure for constructing the pedal curves are as follows:
Establish the pedal point.
Construct a tangent line to the curve, in this case the parabola.
Construct a line perpendicular to the tangent line through the pedal point.
Construct the intersection of these two lines.
Trace the intersection.
Animate the tangent line.
Here is the link to the construction files for the pedal curves.
Negative Pedal Curves
The pedal curve of a line appears to trace out a parabola.
The pedal curve to the right strophoid traces out a parabola opening to the right. The pedal point is on the origin.
The pedal curve to the trisectrix of Maclaurin traces out a parabola opening up to the left. The pedal point is on the origin.
The pedal curve to the cissoid of Diocles appears to trace out a parabola opening to the left. The pedal point is on the origin.
Here is a link to the Geometer’s Sketchpad construction file.
Problem Three:
Hypocycloid of 3 cusps (deltoid), along with its pedal curve traced out in red. The pedal curve wrt was constructed by using the origin as the pedal point, and then constructing a line perpendicular to the line tangent to the curve through the pedal point. When we trace the intersection of these two lines we construct the pedal curve.
The negative pedal curve wrt is constructed by tracing out the same tangent line, also in blue, as this is also the line perpendicular to the line segment from the pedal point O to the point P on the curve. We see that the negative pedal curve reconstructs the original hypocycloid. Thus the negative pedal curve and the pedal curve must be inverse structures.
Here is the hypocycloid of 4 cusps, its pedal curve traced out in blue and the negative pedal curve traced in green.
Here is the hypocycloid of 5 cusps with the pedal curve wrt traced in blue and the negative pedal curve wrt traced in orange. We can easily see the relationship between the pedal curve and negative pedal curve.
In this graph we have included the related epicycloids traced outside the hypocycloid in blue are a family of curves. Notice that the number of cusps is the same for both graphs and they “meet” at the same points. Clearly there exists a relationship between the pedal curve of a hypocycloid and the pedal curve of its related epicycloid. The hypocycloid is related to the idea of picking a point on a circle and then rolling that circle along the inside of a larger circle, without slipping, while tracing the chosen point. Anyone who has ever played with a Spirograph as a child has drawn a hypocycloid. Then the related epicylcoid could be constructed by tracing the same point on the same circle only rolling it along the outside of the circle, starting at the same point.
The equation of the general epi/hypocycloid: {(a + b)*Cos[t] + b*Cos[(a + b)/b*t], (a + b)*Sin[t] + b*Sin[(a + b)/b*t]}. This parameterization has the property that when b is positive, it generates an epicycloid. When b is negative, it is a hypocycloid. Thus, a signed parameter b uniquely defines an epi/hypocycloid.[1]