Question

I need to draw a line in my website, actually a curve representing a third degree polynom.
What is the easiest way of finding a third degree equation that fits two points with given slopes in javascript?

Find the third degree equation for(or find the coeffecient a,b,c,d in general formula ax^3+bx^2+cx+d = y):

startX, startY, startSlope

endX, endY, endSlope

Answer 1

Essentially this is just straightforward math.

You’ve got an unknown expression y=ax³+bx²+cx+d. You can drop quite a few terms by defining x’ = (x-startX)/endX (i.e. startX’ = 0, endX’ = 1). You’ll also have to scale the slopes; startSlope’ = startSlope * 1/(endX-startX).

From this it follows that d’ = startY. That’s your first free parameter.

Next, note that the slope is trivially obtained by differentiation. dy/dx’ = 3a’x’²+2b’x’+c’. Therefore, c’ is just startSlope’.

a’ and b’ take a pair of equations: endY = a’+b’+c’+d’, and endSlope = 3a’+2b’+c’+d’. Therefore a’ = endSlope’ – 2*endY, and b’ = 3*endY – endSlope’.