I'm trying to write a method that interpolates from 0 to x (position of an object in one dimension) over time using acceleration at the beginning and deceleration at the end (ease out / ease in) with the only constraints that the total time is provided, as well as the duration of the acceleration and deceleration. the motion should replicate the inertia effect and I'm considering a Hermite curve for the non-linear portions.
double Interpolate(
double timeToAccel, double timeCruising, double timeToDecel,
double finalPosition,
double currentTime)
{
//...
}
Can someone point me out to a portion of code that does that? I don't know how to integrate the Hermite curve, hence don't know how much I'll move in the accelerating portion or in the decelerating portion, and in turn I can't figure out what will be the speed in the linear portion.
Thanks.
Some reference to illustrate my question.
Edit:
Edit: Roman and Bob10 have provided full working solutions. I implemented the code from Roman. Thanks to you both, guys! I appreciate your perfect support and your detailed solutions, you saved me long searches and trials.
Hermite curves are simple to calculate but also more powerful. They are used to well interpolate between key points. These four vectors are basically multiplied with four Hermite basis functions h1(s), h2(s), h3(s) and,h4(s) and added together.
Hermite cubic curve is also known as parametric cubic curve, and cubic spline. This curve is used to interpolate given data points that result in a synthetic curve, but not a free form, unlike the Bezier and B-spline curves, The most commonly used cubic spline is a three-dimensional planar curve (not twisted).
A piecewise cubic hermite curve is a curve that is represented with four degrees of freedom. Two degrees of freedom are defined as the positions of the two end points of the curve. The other two degrees of freedom are defined as the tangents to the endpoints of the curve.
In numerical analysis, a cubic Hermite spline or cubic Hermite interpolator is a spline where each piece is a third-degree polynomial specified in Hermite form, that is, by its values and first derivatives at the end points of the corresponding domain interval.
First, let's make a cubic hermite spline function:
/*
t - in interval <0..1>
p0 - Start position
p1 - End position
m0 - Start tangent
m1 - End tangent
*/
double CubicHermite(double t, double p0, double p1, double m0, double m1) {
t2 = t*t;
t3 = t2*t;
return (2*t3 - 3*t2 + 1)*p0 + (t3-2*t2+t)*m0 + (-2*t3+3*t2)*p1 + (t3-t2)*m1;
}
Now your task is to calculate the p0, p1, m0 and m1 for both ease-in and ease-out portions. Let us add a few variables to make the math a bit easier to write:
double Interpolate(
double timeToAccel, double timeCruising, double timeToDecel,
double finalPosition,
double currentTime) {
double t1 = timeToAccel;
double t2 = timeCruising;
double t3 = timeToDecel;
double x = finalPosition;
double t = currentTime;
We need to specify where should the object be when it stops accelerating and starts decelerating. You can specify these however you please and still produce a smooth movement, however, we would like a somewhat "natural" solution.
Let's assume that the cruising speed is v
. During crusing, the object travels distance x2 = v * t2
. Now, when the object accelerates from 0 to speed v, it travels distance x1 = v * t1 / 2
. Same for deceleration x3 = v * t3 / 2
. Put all together:
x1 + x2 + x3 = x
v * t1 / 2 + v * t2 + v * t3 / 2 = x
From that we can calculate our speed and the distances:
double v = x / (t1/2 + t2 + t3/2);
double x1 = v * t1 / 2;
double x2 = v * t2;
double x3 = v * t3 / 2;
And now that we know everything, we just feed it into our cubic hermite spline interpolator
if(t <= t1) {
// Acceleration
return CubicHermite(t/t1, 0, x1, 0, v*t1);
} else if(t <= t1+t2) {
// Cruising
return x1 + x2 * (t-t1) / t2;
} else {
// Deceleration
return CubicHermite((t-t1-t2)/t3, x1+x2, x, v*t3, 0);
}
}
I tested this in Excel, here's the equivalent VBA code to play with. There are some divisions by zero for boundary conditions, I leave fix to this as an excercise to the reader
Public Function CubicHermite(t As Double, p0 As Double, p1 As Double, _
m0 As Double, m1 As Double) As Double
t2 = t * t
t3 = t2 * t
CubicHermite = (2 * t3 - 3 * t2 + 1) * p0 + _
(t3 - 2 * t2 + t) * m0 + (-2 * t3 + 3 * t2) * p1 + (t3 - t2) * m1
End Function
Public Function Interpolate(t1 As Double, t2 As Double, t3 As Double, _
x As Double, t As Double) As Double
Dim x1 As Double, x2 As Double, x3 As Double
v = x / (t1 / 2 + t2 + t3 / 2)
x1 = v * t1 / 2
x2 = v * t2
x3 = v * t3 / 2
If (t <= t1) Then
Interpolate = CubicHermite(t / t1, 0, x1, 0, v*t1)
ElseIf t <= t1 + t2 Then
Interpolate = x1 + x2 * (t - t1) / t2
Else
Interpolate = CubicHermite((t-t1-t2)/t3, x1+x2, x, v*t3, 0)
End If
End Function
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