Components of a Parametric 3D Curve: Tangent, Normal, Binormal, and Osculating P
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This GeoGebra example visualizes the differential geometry components of a parametric 3D curve at a moving point, including the unit tangent vector, unit normal vector, binormal vector, normal plane, osculating plane, and curvature. It demonstrates the Frenet-Serret frame and how these vectors and planes change along the curve. The construction uses the CurveCartesian command to define the parametric curve r(t), and Translate and Circle commands to draw vector arrows and the osculating circle. A slider controls the parameter t, updating the point P, tangent T, normal N, binormal B, normal plane np, and osculating plane op in real time. This is ideal for university-level calculus or differential geometry courses, as well as advanced high school students studying vector functions and space curves, providing an interactive way to grasp local properties of curves in 3D.
GeoGebra Commands & Parameters
CurveCartesian(rx(t), ry(t), rz(t), t, minTime, maxTime) → rTranslate(Vector[(rx'(t), ry'(t), rz'(t)) / sqrt(rx'(t)^(2) + ry'(t)^(2) + rz'(t)^(2))], P) → TTranslate(Vector[(rx''(t), ry''(t), rz''(t)) / sqrt(rx''(t)^(2) + ry''(t)^(2) + rz''(t)^(2))], P) → NTranslate(Vector[T ⊗ N], P) → BCircle(P + (N * sqrt(rx'(t)^(2) + ry'(t)^(2) + rz'(t)^(2)) / sqrt(rx''(t)^(2) + ry''(t)^(2) + rz''(t)^(2))), sqrt(rx'(t)^(2) + ry'(t)^(2) + rz'(t)^(2)) / sqrt(rx''(t)^(2) + ry''(t)^(2) + rz''(t)^(2)), op) → c
AI Drawing Prompt Reference
Draw a parametric 3D curve and show its tangent vector, normal vector, binormal vector, normal plane, and osculating plane at a point. First define the curve r(t), then calculate the tangent vector T, normal vector N, and binormal vector B. Next, draw the normal plane and osculating plane, and finally demonstrate how these vectors and planes change with parameter t.
The prompt above was compiled from the original drawing commands and is for reference only.