Space Curve

A curve in space.

Unit Tangent Vector

$$\vec{T}(t)=\frac{{\vec{r}}^{\prime }(t)}{\Vert {\vec{r}}^{\prime }(t)\Vert },\vec{T}(s)={\vec{r}}^{\prime }(s)\left(\Vert {\vec{r}}^{\prime }(s)\Vert =1\right)$$

where $s$ is Arc Length, $t$ is any parameter

Unit Normal Vector

Also known as “principal unit normal”

$$\vec{N}(s)=\frac{{\vec{T}}^{\prime }(s)}{\Vert {\vec{T}}^{\prime }(s)\Vert }\vec{N}(t)=\frac{{\vec{T}}^{\prime }(t)}{\Vert {\vec{T}}^{\prime }(t)\Vert }$$

where $s$ is Arc Length, $t$ is any parameter

Unit Binormal Vector

$$\vec{B}(t)=\vec{T}(t)\times \vec{N}(t)$$

Misc

• Unit normal vector and unit tangent vector are always perpendicular to each other.
• Unit normal vector is always normal to the curve. Unit tangent vector is always tangent to the curve, so their perpness is a natural consequence.

Curvature

The curvature of a Space Curve

$$\kappa (s)=\Vert {\vec{T}}^{\prime }(s)\Vert \kappa (t)=\frac{\Vert {\vec{T}}^{\prime }(t)\Vert }{\Vert {\vec{r}}^{\prime }(t)\Vert }=\frac{\Vert \overrightarrow{r^{\prime }}(t)\times {\vec{r}}^{\prime \prime }(t)\Vert }{\Vert {\vec{r}}^{\prime }(t){\Vert }^3}$$

where $\vec{T}(s)$ is the Unit Tangent Vector and $s$ is Arc Length

Osculating Circle

The circle whose curvature is the same as a Space Curve at a particular point, and whose radius $R$ is reciprocal to the curvature at that point. That is to say:

$$\kappa =\frac{1}{R}$$Link to original