Implicit Curves from Definitions in CAS
This activity belongs to the GeoGebra book GeoGebra Principia.
Implicit Curves from Definitions in CAS
A parabola can be defined as the locus of points in the plane equidistant from a line (directrix) and an external point (focus). Locating one point (the vertex) is easy, but how do we locate the others?
With GeoGebra, we can create a free point to explore the situation and mark those positions where both distances are equal. It's quite instructive but, after several exercises, it becomes tedious.
Alternatively, we can construct a generic point that defines the locus, but this construction will only work for this case or similar cases.
We can also create the implicit curve by defining an arbitrary point X(x,y) in the CAS View:
X:= (x, y)
the distance from X to the focus F:
XF(x,y):= Distance(X, F)
the distance from X to the directrix r:
Xr(x,y):= Distance(X, r)
and by equating both distances:
XF – Xr = 0
GeoGebra uses numerical algorithms to create this implicit curve, so small errors or omissions may appear in some cases.
- Note: At least for now, GeoGebra GeoGebra does not represent equations of this type in three variables. That is, it recognizes x² + y² + z² = 16 as a sphere, but it does not recognize the equivalent equation (sqrt(x² + y² + z²))² = 16 as such.
Author of the construction of GeoGebra: Rafael Losada.