Fitting parabolas and other conic sections to a set of points

William Henney - DAWGI Meeting - 2024-02-28

Abstract

I will demonstrate a method for finding the orientation of the symmetry axis of an observed bow shock or other curved emission arc. The method consists in fitting a parabola (or ellipse, or hyperbola) to a set of points. A simple trick allows a suitable objective function to be defined without needing to explicitly calculate the distance of the points from the curve. The objective function can be minimized to find the best-fit conic section, and MCMC can be used to check for the presence of multiple local minima.

Ajuste de parábolas y otras secciones cónicas a un conjunto de puntos

Voy a demostrar un método para encontrar la orientación del eje de simetría de un arco de emisión observado. El método consiste en ajustar una parábola (o elipse, o hipérbola) a un conjunto de puntos. Un truco simple permite definir una función objetivo adecuada sin necesidad de calcular explícitamente la distancia de los puntos a la curva. La función objetivo se puede minimizar para encontrar la mejor sección cónica de ajuste, y MCMC se puede utilizar para comprobar la presencia de múltiples mínimos locales.

Python package

• https://github.com/div-B-equals-0/confit

Animation

Individual slides

Title page

Fitting conic sections to data: Parabolae, ellipses, and hyperbolae

What?

From image … to points … to fitted curve

I can give another talk later on Stage 1

Why?

Curved emission arcs in H II regions

Also in infrared …

And radio wavelengths too

… and not just in Orion

By fitting a conic section, we can measure

For instance, we may wish to know the orientation in order to identify the exciting source, or to compare with the magnetic field direction, etc

But what if the shape is not a conic section?

How?

Algebraic distance versus geometric distance

Disadvantages of algebraic distance

No clear intuitive interpretation of A, B, C, etc

Geometric Euclidean distance

Yes there is a simpler way

This can be generalized to other forms of conics

Focal radius r and directrix distance d are trivial to calculate

If this is so clever, why did nobody else do it?

Implementation: my confit python package

lmfit is for when astropy.modeling isn’t enough

The objective function calculates a vector of residuals, for which the Minimizer tries to minimize the sum of squares

By default it uses a Levenberg-Marquardt algorithm

Examples

A simple curve with 7 points

General conic versus parabola

Unwanted spurious solutions

Back to normal!

With freely varying eccentricity we still have issues

This is a case where the covariance matrix underestimates the true uncertainties

Better to use a parabola when the number of data points is small

Test on real data

A success

But some edge cases still need to be finessed

Conclusions

Fitting conic sections can be useful, easy, and fun