https://github.com/maxkapur/param_tools
Integrate and generate random points over parametric curves and surfaces.
https://github.com/maxkapur/param_tools
integration numerical-methods numpy python
Last synced: 7 months ago
JSON representation
Integrate and generate random points over parametric curves and surfaces.
- Host: GitHub
- URL: https://github.com/maxkapur/param_tools
- Owner: maxkapur
- Created: 2020-06-14T01:31:56.000Z (over 5 years ago)
- Default Branch: master
- Last Pushed: 2020-06-18T03:57:01.000Z (over 5 years ago)
- Last Synced: 2025-03-28T11:21:16.499Z (10 months ago)
- Topics: integration, numerical-methods, numpy, python
- Language: Python
- Size: 1.62 MB
- Stars: 7
- Watchers: 1
- Forks: 0
- Open Issues: 0
-
Metadata Files:
- Readme: readme.md
Awesome Lists containing this project
README
This module contains several functions useful for evaluating integrals and generating random points on parametric curves and surfaces.
Author: [Max Kapur](maxkapur.com)
### Examples
Generate uniformly distributed random points along a parametric curve:
Generate uniformly distributed random points on a parametric surface:
Please see the Jupyter notebook `examples.ipynb` or its HTML counterpart for workflow examples.
### Calculating arc lengths and surface areas
`arc_cumulator()` and `surface_cumulator()` take data coordinates as input and return the arc length or area accumulated at each point. If the parameterizing function is known, the convenience functions `arc_length()` and `surface_area()` return the total arc length or surface area over the given region. The precision arguments to `arc_length()` and `surface_area()` determine how many subintervals will be evaluated.
### Generating random data
The functions `r_arc()` and `r_surface()`, and their counterparts `r_arc_from_data()` and `r_surface_from_data()`, let you generate random points that are uniformly distributed with respect to the arc length of a parametric curve or the area of a parametric surface.
For a curve given by a parametric function `func(t)` for a certain range of t-values, use `r_arc(n, func, t0, t1)`. For a surface given by a parametric function `func(t,u)` for a range of t- and u-values, use `r_surface(n, func, t0, t1, u0, u1)`. The parameter space must be rectangular.
For a curve given by a list of point coordinates and (optionally) corresponding t-values, use `r_arc_from_data(n, t, coords)`. For a curve given by a grid of point coordinates and (optionally) corresponding t- and u-values, use `r_surface_from_data(n, t, u, coords)`, where the input is best formatted like a numpy meshgrid. In these functions, Scipy's interpolation functions are used to return points on the surface; the interpolation method may be controlled by the `kind` parameter.
Mathematically, the procedure for sampling a random point on an arc given by `f(t)` on `[t0, t1]` is as follows:
1. Numerically compute the total arc length `P`.
2. Pick a random number `s` on `[0, P)`.
3. Determine, by interpolation, `t` such that the arc length on `[t0, t]` is `s`.
4. Return `f(t)`.
For a surface given by `f(t, u)`, the process is similar:
1. Numerically compute the total area `A`.
2. Independently pick two random numbers `S1` and `S2` on `[0, A)`.
3. Determine, by interpolation, t such that the surface area on `[t0, t], [u0, u1]` is `S1`.
4. Determine, by interpolation, u such that the surface area on `[t0, t1], [u0, u]` is `S2`.
5. Return `f(t, u)`.
### Acknowledgements
The following Stack Exchange discussions were of assistance in creating this module:
- [Generate random points on perimeter of ellipse](https://math.stackexchange.com/questions/3710402/generate-random-points-on-perimeter-of-ellipse/3718774#3718774) (Math SE)
- [Generate random points on perimeter of ellipse](https://codereview.stackexchange.com/questions/243590/generate-random-points-on-perimeter-of-ellipse/243697?noredirect=1#comment478326_243697) (Code Review SE)