Find inflection point of parametric cubic I'm familiar with turning points and inflection points for &quot;normal&quot; graphs (i.e. those that relate y and x) but how would I get an inflection point for a graph where each dimension has a separate equation?
In my case, given a parametric cubic curve $$\vec r(t) = \vec At^3 + \vec Bt^2 + \vec Ct + \vec D$$ , how would I determine the inflection point? Also, what does this point actually represent? Change in concavity? I am struggling to conceptualize this in 3D Vector space.
Thanks!
A parametrized curve has a tangent vector $$\overrightarrow{T} = \frac{\overrightarrow{dr}/dt}{|\overrightarrow{dr}/dt|} $$ divided by its length.
The curvature $\kappa$ (a scalar) is defined as $$\kappa =\frac{|\overrightarrow{dT}/dt|}{|\overrightarrow{dr}/dt|}$$
The unit normal vector is $$\overrightarrow{N} = \frac{\overrightarrow{dT}/dt}{\vert{\overrightarrow{dT}/dt}\vert} $$
An inflection point would be where the curvature is zero.
A parametrized curve has a tangent vector $$\overrightarrow{T} = \frac{\overrightarrow{dr}/dt}{|\overrightarrow{dr}/dt|} $$ divided by its length.
The curvature $\kappa$ (a scalar) is defined as $$\kappa =\frac{|\overrightarrow{dT}/dt|}{|\overrightarrow{dr}/dt|}$$
The unit normal vector is $$\overrightarrow{N} = \frac{\overrightarrow{dT}/dt}{\vert{\overrightarrow{dT}/dt}\vert} $$
An inflection point would be where the curvature is zero.