Speed and Curvature in Parametric Form — Theory
1. Parametric vector curves
A parametric curve can be written as a vector function:
\[
\mathbf r(t)=
\langle x(t),y(t),z(t)\rangle.
\]
For a plane curve, use \(z(t)=0\). For a space curve such as a helix, use all three components.
2. Velocity vector
The velocity vector is the derivative of position:
\[
\mathbf v(t)=\mathbf r'(t)
=
\langle x'(t),y'(t),z'(t)\rangle.
\]
It points in the instantaneous direction of motion along the curve.
3. Speed
Speed is the magnitude of the velocity vector:
\[
|\mathbf v(t)|
=
\sqrt{x'(t)^2+y'(t)^2+z'(t)^2}.
\]
If \(x,y,z\) are measured in meters and \(t\) is measured in seconds, then speed is measured in
\(\mathrm{m/s}\).
4. Acceleration vector
Acceleration is the derivative of velocity:
\[
\mathbf a(t)=\mathbf r''(t)
=
\langle x''(t),y''(t),z''(t)\rangle.
\]
Acceleration measures how the velocity vector changes.
5. Unit tangent vector
The unit tangent vector gives the direction of motion without the speed scale:
\[
\mathbf T(t)=
\frac{\mathbf r'(t)}{\|\mathbf r'(t)\|}.
\]
This vector is defined only when the speed is nonzero.
6. Curvature
Curvature measures how sharply the curve bends. For a space curve, the standard formula is
\[
\kappa(t)=
\frac{\|\mathbf r'(t)\times\mathbf r''(t)\|}
{\|\mathbf r'(t)\|^3}.
\]
A larger \(\kappa\) means sharper bending. A smaller \(\kappa\) means a flatter or gentler curve.
7. Plane-curve curvature
For a plane curve \(\mathbf r(t)=\langle x(t),y(t)\rangle\), curvature can be written as
\[
\kappa(t)=
\frac{|x'(t)y''(t)-y'(t)x''(t)|}
{\left(x'(t)^2+y'(t)^2\right)^{3/2}}.
\]
This is the two-dimensional version of the cross-product formula.
8. Radius of curvature
The radius of curvature is the reciprocal of curvature:
\[
\rho(t)=\frac{1}{\kappa(t)}.
\]
If the curve bends sharply, \(\kappa\) is large and \(\rho\) is small. If the curve bends gently,
\(\kappa\) is small and \(\rho\) is large.
9. Osculating circle
The osculating circle is the circle that best matches the curve near a point. Its radius is
\[
\rho=\frac{1}{\kappa}.
\]
For a plane curve, the osculating circle is drawn in the curve plane. For a 3D curve, the calculator
shows the circle in the \(x\)-\(y\) projection while computing curvature with the full 3D formula.
10. Tangential and normal acceleration
Acceleration can be split into tangential and normal parts:
\[
\mathbf a
=
a_T\mathbf T+a_N\mathbf N.
\]
The tangential component changes speed. The normal component changes direction.
\[
a_T=\frac{\mathbf v\cdot\mathbf a}{\|\mathbf v\|}.
\]
11. Example: helix
A common space curve is the helix
\[
\mathbf r(t)=
\langle \cos t,\sin t,t\rangle.
\]
Then
\[
\mathbf r'(t)=
\langle -\sin t,\cos t,1\rangle,
\]
and the speed is constant:
\[
\|\mathbf r'(t)\|=\sqrt{2}.
\]
12. Example: cycloid
A cycloid can be written as
\[
x=t-\sin t,
\qquad
y=1-\cos t.
\]
It has sharp behavior near cusp points where the speed can become very small.
Curvature formulas become sensitive near those points.
13. Units on graphs
If \(x,y,z\) are measured in meters and \(t\) is measured in seconds, then the curve graph should show
\[
x\;(\mathrm{m}),
\qquad
y\;(\mathrm{m}).
\]
The quantity graph should also show its units. For example:
\[
|\mathbf v|\;(\mathrm{m/s}),
\qquad
\kappa\;(\mathrm{1/m}).
\]
15. Common mistakes
- Forgetting that speed is a magnitude: speed is not the same as the velocity vector.
- Using only \(x'\) or \(y'\): speed uses all coordinate derivatives.
- Dividing by zero speed: curvature is undefined when \(\|\mathbf r'(t)\|=0\).
- Confusing curvature and radius: \(\rho=1/\kappa\).
- Ignoring units: curvature has inverse distance units, while radius of curvature has distance units.
- Using a 2D formula for a 3D curve: use the cross-product formula for space curves.
