# LocusEquation Command

##### Command Categories (All commands)

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.
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