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To solve it in C/C++, you could either implement the Gauss algo (see also the Numerical Recipes book, it is available online), or use Linear Algebra libraries, such as Eigen, or others. Remark: the approach is the same regardless if the point (x4, y4) lies within the triangle (x1, y1), (x2, y2), (x3, y3) , or not.
I'm trying to come up with a simple and efficient way to create a smooth surface which intersects a number of given "sample" points. For any X,Y point on the surface, I identify up to 4 sample points in each of the 4 directions (the next higher and lower points on the X, and then the Y axes).
Trilinear interpolation is a method of multivariate interpolation on a 3-dimensional regular grid. It approximates the value of a function at an intermediate point within the local axial rectangular prism linearly, using function data on the lattice points.
So, now we have (x, y, z) (x, y, z) as a function of t t, so we have a parametric space curve. To do 3D spline interpolation using Matlab functions, see here. A better reference is this web site. Bezier curves are also easy to extend to 3D.
This MATLAB function returns interpolated values of a function of n variables at specific query points using linear interpolation.
The plane where the curve lies, a 2D vector space. The space of cubic polynomials, a 4D space. Don’t be confused! The 2D control points can be replaced by 3D points – this yields space curves. The math stays the same, just add z(t). The cubic basis can be extended to higher-order polynomials.
• Interpolating in the space of 3D vectors is well behaved • Simple computation: interpolate linearly and normalize – this is what we do all the time, e.g. with normals for fragment shading – but for far-apart endpoints the speed is uneven (faster towards the middle) • For constant speed: spherical linear interpolation – build basis {v
This MATLAB function returns interpolated values of a function of three variables at specific query points using linear interpolation.
As a first approach, sample points at increasing values of t and accumulate the chord lengths. Inversion will imply dichotomic search and inverse linear interpolation. Additional suggestion: for good accuracy, you can implement a good numerical integration scheme, giving you a number of (t, s) values along the curve.
The trivariate interpolation allows obtaining values at arbitrary points in a 3D space of a function defined on a grid. This method performs a bilinear interpolation in 2D space by considering the axes of longitude and latitude of the grid, then performs a linear interpolation in the third dimension. Its interface is similar to the bivariate ...