C＃/ Unity3D中的分段线性整数曲线插值

Question

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 wrt arrival time, ie 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

Answer 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