# LocusEquation Command

From GeoGebra Manual

- 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.