Focus Vertex Paths for Rolling Parabola
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This example demonstrates the locus of the focus and vertex of a rolling parabola. It involves core mathematical concepts like parabola, similar triangles, geometric mean, right triangles, and division. The construction utilizes GeoGebra commands such as Focus, OrthogonalLine, Parabola, Locus, Distance, CurveCartesian, and objects like p, c, t, tp, F', F, V', L, dir, d, Fpath, Vpath, ShF, ShV, Directios, tEq0, togT, fc, FV', Vp, VertexLocEq, fEq, vEq, tis, P, s, P', Fp, FocLocParamEq, FocLocEq. Suitable for high school and above students to explore dynamic geometric properties and locus problems.
GeoGebra Commands & Parameters
Focus(c) → F,OrthogonalLine(L, Segment[F', L]) → dirParabola(F', dir) → dLocus(F', t) → FpathLocus(V', t) → VpathDistance(F, V') → FV'CurveCartesian((p * log((sqrt(t^(2) + (4 * p^(2))) + t) / ((2 * p)))) - (p * t / sqrt(t^(2) + (4 * p^(2)))), t^(2) / ((2 * sqrt(t^(2) + (4 * p^(2))))), t, -35, 35) → VpCurveCartesian((p * log((sqrt(t^(2) + (4 * p^(2))) + t) / ((2 * p)))), sqrt(t^(2) + (4 * p^(2))) / 2, t, -35, 35) → Fp
AI Drawing Prompt Reference
Draw a rolling parabola and show the paths traced by its focus and vertex. Start with the base parabola and its focus. Construct a line perpendicular to the segment joining the focus and the vertex, use it as the directrix to form a new parabola. As the rolling parameter changes, trace the locus of the focus and vertex, yielding two smooth curves. Also plot the parametric curves for the focus and vertex paths, and mark the distance between the focus and the rolling vertex to illustrate the geometric relationship.
The prompt above was compiled from the original drawing commands and is for reference only.