Piecewise linear integer curve interpolation in C#/Unity3D

问题

Is there a simple, efficient way to implement a piecewise linear integer-to-integer curve interpolation in C# (for Unity3D, if it matters) ?
Details are as follows:

The piecewise linear curve representation has to be built over time. The first interpolation request comes before we have all data points
The curve is strictly monotonous
The first point is always (0, 0)
The data points' first coordinates are also strictly monotonous w.r.t arrival time, i.e. the points are naturally ordered by their first coordinate.
The data points are not in ranges that would cause cause overflow problems for 4-byte integers
The output does not have to be 100% accurate, so rounding errors are not an issue.

In C++, I would do something like this:

#include <algorithm>
#include <vector>
#include <cassert>

using namespace std;

typedef pair<int, int> tDataPoint;
typedef vector<tDataPoint> tPLC;

void appendData(tPLC& curve, const tDataPoint& point) {
  assert(curve.empty() || curve.back().first < point.first);
  curve.push_back(point);
}

int interpolate(const tPLC& curve, int cursor) {
  assert(!curve.empty());
  int result = 0;  
  // below zero, the value is a constant 0
  if (cursor > 0) {
    // find the first data point above the cursor
    const auto upper = upper_bound(begin(curve), end(curve), cursor);
    // above the last data point, the value is a constant 0
    if (upper == end(curve)) {
      result = curve.back().second;
    } else {
      // get the point below or equal to the cursor
      const auto lower = upper - 1;
      // lerp between
      float linear = float((cursor - lower.first) * (upper.second - lower.second)) / (upper.first - lower.first);
      result = lower.second + int(linear);
    }
  }
  return result;
}

I can see how I could do something that work sort of like this in C#, but nothing as concise or efficient. Any help will be appreciated.

EDIT: I do not need to be more accurate, and am perfectly happy with piecewise linear interpolation, so better interpolation quality is not my problem here.
What I am looking for is an efficient, concise way of doing this. By efficient, I mean things like: relying on the fact that the data points are naturally ordered to be able to use binary search to find the proper segment

回答1:

I would use this interpolation cubic:

x=a0+a1*t+a2*t*t+a3*t*t*t
y=b0+b1*t+b2*t*t+b3*t*t*t

where a0..a3 are computed like this:

d1=0.5*(p2.x-p0.x);
d2=0.5*(p3.x-p1.x);
a0=p1.x;
a1=d1;
a2=(3.0*(p2.x-p1.x))-(2.0*d1)-d2;
a3=d1+d2+(2.0*(-p2.x+p1.x));

b0 .. b3 are computed in same way but use y coordinates of course
p0..p3 are control points for cubic interpolation curve
t = < 0.0 , 1.0 > is curve parameter from p1 to p2

This ensures that position and first derivation is continuous (c1). If you want to do this on integer math then just scale ai,bi ant t accordingly. You can also add as many dimensions as you need in the same manner

Now you need some parameter to go through your interpolation points for example u = <0 , N-1>

p(0..N-1) are your control points list
u = 0 means start point p(0)
u = N-1 means end point p(N-1)
P0..P3 are control points used for interpolation

So you need to compute t and select which points to use for interpolation

    double t=u-floor(u); // fractional part between control points
    int i=floor(u);       // integer part points to starting control point used
         if (i<1)     { P0=p(  0),P1=p(  0),P2=p(  1),P3=p(  2); }               // handle start edge case
    else if (i==N-1) { P0=p(N-2),P1=p(N-1),P2=p(N-1),P3=p(N-1); }  // handle end edge case
    else if (i>=N-2) { P0=p(N-3),P1=p(N-2),P2=p(N-1),P3=p(N-1); }  // handle end edge case
    else              { P0=p(i-1),P1=p(i  ),P2=p(i+1),P3=p(i+2); }

    (x,y) = interpolation (P0,P1,P2,P3,t);

If you want to do this on integer math then just scale u,t accordingly. If N<3 then use linear interpolation ... or duplicate end points until N>=3

[edit1] linear interpolation approach

struct pnt { int x,y; };

pnt interpolate (pnt *p,int N,int x)
    {
    int i,j;
    pnt p;
    for (j=1,i=N-1;j<i;j<<=1); j>>=1; if (!j) j=1; // this just determine max mask for binary search ... can do it on p[] size change
    for (i=0;j;j>>=1) // binary search by x coordinate output is i as point index with  p[i].x<=x
        {
        i|=j;
        if (i>=N) { i-=j; continue; }
        if (p[i].x==x) break;
        if (p[i].x> x) i-=j;
        }
    p.x=x;
    p.y=p[i].y+((p[i+1].y-p[i].y)*(x-p[i].x)/(p[i+1].x-p[i].x))
    return p;
    }

add edge cases handling like x is out of points bound or point list is too small

来源：https://stackoverflow.com/questions/22803762/piecewise-linear-integer-curve-interpolation-in-c-unity3d

标签

c++

unity3d

interpolation

linear-interpolation