Bezier surface intersection with a plane.

ClsSensorPlaneIntersector.cs

using System;
using System.Collections.Generic;
using System.Linq;
using System.Web;

namespace Kiss.Effigy.code {
    public class ClsSensorPlaneIntersector {
        public Vector Origin;
        public Func<float, float, Vector> Parametric;
        public float Plane;
        public float Epsilon = 0.01f;
        public ClsSensorPlaneIntersector(Vector origin, Func<float, float, Vector> parametric, float elevation) {
            // The first argument is the position of the region's origin.
            // The second argument is a parametric function of two variables:
            // - the first float is an angle, in radians, of rotation.
            // - the second float is the height parameter, ranging from 0 to 1.
            // The third argument is an elevation of intersection.  
            Origin = origin;
            Parametric = parametric;
            Plane = elevation;
        }
        public Boolean FuzzyEqual(float a, float b) {
            return Math.Abs(a - b) <= Epsilon;
        }
        public Boolean FloatInRange(float x, float a, float b) {
            return (a < x && b > x) || (a > x && b < x);
        }
        public Vector FindPointIntersection(Vector origin, Func<float, Vector> surfaceSlice, float elevation, float min, float max) {
            var minE = (float)surfaceSlice(min).k;
            var maxE = (float)surfaceSlice(max).k;
            var mid = (max + min) / 2;
            var midE = (float)surfaceSlice(mid).k;
            if (FuzzyEqual(elevation, midE) || FuzzyEqual(mid, max)) {
                // If a point was found that is sufficiently close to the place, or
                // if the range has been drilled down to a sufficiently small span,
                // return the point of focus.
                return surfaceSlice(mid);
            }
            if (FloatInRange(elevation, minE, midE)) {
                return FindPointIntersection(origin, surfaceSlice, elevation, min, mid);
            } else if(FloatInRange(elevation, midE, maxE)) {
                return FindPointIntersection(origin, surfaceSlice, elevation, mid, max);
            } else {                
                throw new Exception("Could not intersect");
                return null;
            }
        }
        public List<Vector> Intersect() {
            var precision = 20;
            var intersections = new List<Vector>();
            for (var theta = 0f; theta <= Math.PI * 2; theta += ((float)Math.PI / precision)) {
                Func<float, Vector> slice = (float h) => { return Parametric(theta, h);  };
                var intersection = FindPointIntersection(Origin, slice, Plane, 0f, 1f);
                intersections.Add(intersection);
            }
            return intersections;
        }
    }    
}

integrate.md

Func<float, float, Vector> surface = (float theta, float h) => {
    var hp = (h - 0.5f) * 2;
    var R = 10f;
    var r = Math.Sqrt(R*R - hp*hp);
    return new Vector(
        r * Math.Cos(theta),
        r * Math.Sin(theta),
        hp);
};
var placeIntersector = new ClsSensorPlaneIntersector(new Vector(0, 0, 0), surface, -1f);
var ps = placeIntersector.Intersect();