After researching about curves in computer graphics (splines in my case), I have come across something I did not know: Explicit functions like: \$y=x^2+2\$ are not the best way to interpolate between points because you could get 2 or more values for y for the same x. So, parametric functions seem much better for this if we make the commonly used t parameter, the distance from origin, time, etc...
After studying a bit how to convert from explicit to parametric form I have been shocked when I have found in some texts they use these parametric cubic functions to interpolate through points: $$\begin{align} x(t)&=at^3 + bt^2 + ct +d \\ y(t)&=2at^3 + 2bt^2 + 2ct +2d \end{align} $$ but wait... this is the exact same form the explicit function was: $$ y(x)= ax^3 + bx^2 + cx + d $$ From what I have seen for other example functions in some math books the parametric funcions where not the same form that its explicit form counterpart.
Am I missing something?
Examples of the texts I'm refering to could be:
https://www.cs.helsinki.fi/group/goa/mallinnus/curves/curves.html
http://escience.anu.edu.au/lecture/cg/Spline/parametricCubicCurves.en.html
y = f(x)
wherey()
is a cubic function ofx
" or 2) "this is a parametric equationp(t) = (x(t), y(t))
where the functionsx()
andy()
are each cubic functions oft
". I think the definition you've found is using the latter meaning - I can supply coefficients such thaty
is not a cubic function ofx
. \$\endgroup\$