Cubic/Curve Smooth Interpolation in C# [closed]

Question

Below is a cubic interpolation function:

public float Smooth(float start, float end, float amount)
{
    // Clamp to 0-1;
    amount = (amount > 1f) ? 1f : amount;
    amount = (amount < 0f) ? 0f : amount;

    // Cubicly adjust the amount value.
    amount = (amount * amount) * (3f - (2f * amount));

    return (start + ((end - start) * amount));
}

This function will cubically interpolate between the start and end value given an amount between 0.0f - 1.0f. If you were to plot this curve, you'd end up with something like this:

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The cubic function here is:

    amount = (amount * amount) * (3f - (2f * amount));

How do I adjust this to produce two produce tangents in and out?

To produce curves like this: (Linear start to cubic end)

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As one function

and like this as another: (Cubic start to linear end)

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Anyone got any ideas? Thanks in advance.

Answer 1

What you want is a Cubic Hermite Spline:

where p0 is the start point, p1 is the end point, m0 is the start tangent, and m1 is the end tangent

Answer 2

you could have a linear interpolation and a cubic interpolation and interpolate between the two interpolation functions.

ie.

cubic(t) = cubic interpolation
linear(t) = linear interpolation
cubic_to_linear(t) = linear(t)*t + cubic(t)*(1-t)
linear_to_cubic(t) = cubic(t)*t + linear(t)*(1-t)

where t ranges from 0...1

Answer 3

Well, a simple way would be this:

-Expand your function by 2 x and y
-Move 1 to the left and 1 down
Example: f(x) = -2x³+3x²
g(x) = 2 * [-2((x-1)/2)³+3((x-1)/2)²] - 1

Or programmatically (cubical adjusting):

double amountsub1div2 = (amount + 1) / 2;
amount = -4 * amountsub1div2 * amountsub1div2 * amountsub1div2 + 6 * amountsub1div2 * amountsub1div2 - 1;

For the other one, simply leave out the "moving":

g(x) = 2 * [-2(x/2)³+3(x/2)²]

Or programmatically (cubical adjusting):

double amountdiv2 = amount / 2;
amount = -4 * amountdiv2 * amountdiv2 * amountdiv2 + 6 * amountdiv2 * amountdiv2;