Conics

A Python class to classify, manipulate and visualise conic sections. This class is designed to extend the functionality of polynomial_methods

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Conic types are instantiated from a bi-variate polynomial equation by the factory method which is a design pattern in object-oriented programming.

We create objects to represent conic sections without having to specify the type of conic section to create. In this sense, conic_factory.py is a classifier of conic sections.

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Definition

A conic section is a plane algebraic curve of degree two whose coordinates satisfy a quadratic equation in two variables;

$$ Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 $$

with all coefficients $\in \mathbb{R}$ and $A, B, C$ not all zero. This is the general form equation of the conic

The above equation can be written in matrix notation as

$$ \left(\begin{array}{ll} x & y \end{array}\right)\left(\begin{array}{cc} A & B / 2 \\ B / 2 & C \end{array}\right)\left(\begin{array}{l} x \\ y \end{array}\right)+\left(\begin{array}{ll} D & E \end{array}\right)\left(\begin{array}{l} x \\ y \end{array}\right)+F=0 . $$

Conic sections described by this equation can be classified in terms of the discriminant $\Delta = B^2 -4AC$ of the quadratic part of the general equation, $Ax^2 + Bxy + Cy^2$.

The discriminant is $-4\Delta$ where $\Delta$ is the determinant of the quadratic matrix $\textbf{M}$,

$$ \det(\mathbf{M}) = \begin{vmatrix} A & B/2 \\ B/2 & C \end{vmatrix} $$

If the conic is non-degenerate then,

• if $B^2-4 A C<0$, the equation represents an ellipse.
  • if $A=C$ and $B=0$, the equation represents a circle, a special case of an ellipse.
• if $B^2-4 A C=0$, the equation represents a parabola.
• if $B^2-4 A C>0$, the equation represents a hyperbola.
  • if $\tau=A+C=0$, the equation represents a rectangular hyperbola.

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Input Parsing and Processing:

Conic takes a string representation of a polynomial expression. Equalities will not be parsed.

For example;

The equation $x^2+y^2=1$ should be input as 'x^2 + y^2 -1'

The equation $y=x^2$ should be input as 'x^2'

Either the ** or ^ operators are accepted for exponentiation. The * operator is required for multiplication.

Parsing relies on the SymPy library to eliminate fractions and multiply the equation by the LCM thus reducing it to the general form of a conic section with integer coefficients.

The poly_dictionary program then decomposes the equation into a dictionary of the form; {(degree of x, degree of y): coefficient, ...}

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How to create and classify a conic section.

mystery_conic = "x^2/4 + x*y/3 + y^2/9 -2*x + 3*y -1"
conic = Conic.create(mystery_conic)

# Type
In [1]: conic.type
Out[1]: Parabola

# Coefficients
In [2]: conic.coeff
Out[2]: {(2, 0): 9, (1, 1): 12, (1, 0): -72, (0, 2): 4, (0, 1): 108, (0, 0): -36}

The input equation represented a parabola and the output dictionary gives the coefficients of the general form equation.

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Common methods

Once a conic object has been instantiated, get_info() is the first method to call to give an overview of the characteristics along with plot(x_range(min, max), y_range(min, max))

These methods have a common name yet are uniquely defined within each subclass.

get_info() packages a number of useful methods and returns;

• original input string self.__repr__()
• type of conic self.type
• coefficient dictionary self.coeff
• general form equation
• matrices: full and quadratic self.print_matrices
• orientation (if applicable) self.orientation
• axis of symmetry (if applicable) self.axis
• plot self.plot

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Standardisation

Conic sections can be rotated and translated within the $(x, y)$-plane. Should a transformation occur, the state of the conic will be recorded at each stage and can be accessed with self.print_history().

The aim of the transformation is to render the conic to 'standard form' where further attributes can be calculated. Directly calling these attributes is disallowed for a conic not in standard form. The user is directed to first perform a transformation.

Below is a parabola transformed to standard form

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Notebooks

Tutorials demonstrating the methods available to each type of conic section are linked below;

Parabolas

Circles

Ellipse

Hyperbolas

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Installation

Follow these steps to install the Conics package:

Clone the repository:

git clone https://github.com/pineapple-bois/conic.git

Navigate to the cloned repository:

cd conic

Install the package

pip install .

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Further Development

• Adapting poly_dictionary to accept radical coefficients
• Additional parameters to the transformation methods to perform arbitrary translation and rotation in $\mathbb{E}^2$
• Allow scaling and dilation
• Testing is required. I plan to use unittest
• Writing extensive instruction and documentation
• Eliminating all floating-point error accumulation. Currently, the rotation is performed using the floating point rotation angle in radians albeit to a relatively high precision.
• Two conic objects will be allowed to 'interact' (eventually)

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