LocusEquation Command

LocusEquation[ <Locus> ]

Calculates the equation of a Locus and plots this as an Implicit Curve.

LocusEquation[ <Point Creating Locus Line Q>, <Point P> ]

Calculates the equation of a Locus by using inputs tracer point Q and mover point P, and plots this as an Implicit Curve.

Example:

Let us construct a parabola as a locus: Create free Points A and B, and Line d lying through them (this will be the directrix of the parabola). Create free point F for the focus. Now create P on Line d (the mover point), then create line p as a perpendicular line to d through P. Also create line b as perpendicular bisector of Line Segment FP. Finally, point Q (the point creating locus line) is to be created as intersection of Lines p and b. Now LocusEquation[Q,P] will find and plot the exact equation of the locus.

LocusEquation[ <Boolean Expression>, <Free Point> ]

Calculates the locus of the free point such that the boolean condition is satisified.

Example: LocusEquation[AreCollinear[A, B, C],A] for free points A, B, C calculates the set of positions of A that make A, B and C collinear—i.e. a line through B and C.

Notes:

• The Locus must be made from a Point (not from a Slider)
• Works only for a restricted set of geometric loci, i.e. using points, lines, circles, conics. (Rays and line segments will be treated as (infinite) lines.)
• If the locus is too complicated then it will return 'undefined'.
• If there is no locus then the implicit curve is the empty set 0=1.
• If the locus is the whole plane then the implicit curve is the equation 0=0.
• The calculation is done using Gröbner bases, so sometimes extra branches of the curve will appear that were not in the original locus.
• Further informations and examples on geogebra.org. A collection of implicit locus examples is also available.
• See also Locus command and GeoGebra Automated Reasoning Tools: A Tutorial.