pyConics 1.0.1

Handle points, lines and conics by using algebraic geometry.

pyConics

pyConics handles points, lines and conics by using algebraic geometry and homogeneous coordinates (projective geometry).

An overview

pyConics is a personal project that I have been working on since 2020. Its main goal is to provide a tool based on algebraic geometry and linear algebra for manipulating two-dimensional geometric forms such as points, lines, conics and pencil of conics.

It is worth noting that all of the geometric forms in pyConics lie in a Projective space, therefore they are represented by homogeneous coordinates and/or by an homogeneous system of linear equations.

When the pyConics package is finished, it will serve as a basis for the development of a much larger project that consists of solving the problem of multilateration (MLAT), also known as hyperbolic positioning. In other words, it is a method of locating an object by getting the time difference of arrival (TDOA) of a signal emitted from the object to three or more receivers/stations.

Additionally (version 0.2.6), this package can be used by high school and undergraduate students to solve two-dimensional Cartesian geometry problems with points and lines, such as:

1. find a line that passes through two given points.
2. find the point of intersection of two given lines.
3. given a point and a straight line, find a line that crosses this point and is perpendicular to the given line.

In version 1.0.0, besides you being able to solve the problems as above, you will also be able to graph these results, allowing you to illustrate and insert them into any kind of document.

Installation

From a local directory clones this project:

git clone https://github.com/osowsky/pyConics.git (by using http)
or
git clone [e-mail]:osowsky/pyConics.git (by using ssh)

You can install this package, as well:

pip install pyConics

NOTE:
pyConics has been tested on Windows 10 and Elementary 6 OSes. If something is going wrong with other OS, please reach me out so that I can fix your problem.

Usage

pyConics can be used to handle points, lines and conics by using algebraic geometry and homogeneous coordinates.

WARNING:
To avoid any conflict name, from the version 1.0.0 onwards, the Point and Line classes had their names changed to CPoint and CLine, respectively. For this reason, all of the examples in this file, README.md, had to be updated, renaming these classes with their new names. I apologize for that.

Working with points (v0.2.6 onwards)

The representation in homogeneous coordinates of a Cartesian point $p = (\enspace\alpha,\enspace\beta\enspace)$, where $\alpha,\enspace\beta\enspace\in\enspace\mathcal{R}$, is given by the following vector in $\mathcal{R}^3$:

$$ p=\left[\begin{array}{cc} \enspace\alpha & \beta & 1.0\enspace \end{array}\right]^{T} $$

If you want to represent a point $p$ at infinity, you must define a vector as follows:

$$ p=\left[\begin{array}{cc} \enspace\alpha & \beta & 0.0\enspace \end{array}\right]^{T}, $$

where $\enspace\alpha,\enspace\beta\enspace\in\enspace\mathcal{R}$

• How to work with points in pyConics.

    from pyConics import CPoint

    p1 = CPoint( ( 0.0, 1.0 ), 'p1' ) # p1 = ( 0.0, 1.0 ).
    p2 = CPoint( ( 1.0, 1.0 ), 'p2' ) # p2 = ( 1.0, 1.0 ).

    print( p1 ) # -> p1: [0.0000e+00 1.0000e+00 1.0000e+00].
    print( p2 ) # -> p2: [1.0000e+00 1.0000e+00 1.0000e+00].
    print()

    print( f'Are p1 and p2 the same? {p1 == p2}\n' ) # -> False.

    d12 = p1.distance( p2 )
    print( f'Distance from {p1}\nto {p2} is {d12:.4f}.\n' ) # -> d12 = 1.0.

    p3 = CPoint( ( 1, 1, 0 ), 'p3' ) # Point at the infinity.
    print( p3 ) # -> p3: [1.0000e+00 1.0000e+00 0.0000e+00] -> point at the infinity.
    print()

    print( f'Is p3 a point at infinity? {p3.at_infinity()}.\n' ) # -> True.

    d13 = p1.distance( p3 )
    print( f'Distance from {p1} to\n{p3} is {d13:.4f}.\n' ) # -> d13 = Inf.

Working with lines (v0.2.6 onwards)

The representation in homogeneous coordinates of a Cartesian line $l:\beta y=\alpha x + \gamma$, where $\alpha,\enspace\beta,\enspace\gamma\in\enspace\mathcal{R}$, is given by the following vector in $\mathcal{R}^3$:

$$ l=\left[\begin{array}{cc} \enspace\alpha & -\beta & \gamma\enspace \end{array}\right]^{T} $$

The vector above satisfies the following homogeneous expression for straight lines in projective geometry:

$$ l:\left\lbrace\enspace (\enspace x,\enspace y\enspace)\enspace|\enspace \left[\begin{array}{cc} \enspace\alpha & -\beta & \gamma\enspace \end{array}\right]\times \left[\begin{array}{c} \enspace\ x \enspace\ \enspace\ y \enspace\ \enspace\ 1 \enspace\ \end{array}\right]\enspace=\enspace 0 \enspace\right\rbrace $$

If you want to represent a line $l$ at infinity, you must define a vector as follows:

$$ l\enspace=\left[\begin{array}{cc} \enspace 0.0 & 0.0 & \gamma\enspace \end{array}\right]^{T}, $$

where $\enspace\gamma\enspace\in\enspace\mathcal{R}$

• How to work with lines in pyConics.

    from pyConics import CLine

    l1 = CLine( ( 1, -1, 1 ), 'l1' )         # l1: y = x + 1
    l2 = CLine( ( 1.5, -1.5, -1.5 ), 'l2' )  # l2: 1.5y = 1.5x - 1.5
    l3 = CLine( ( -1, -1, 1 ), 'l3' )        # l3: y = -x + 1
    l4 = CLine( ( 2, 2, -2 ), 'l4' )         # l4: 2y = -2x + 2

    print( l1 ) # -> l1: ( x, y ) | [1.0000e+00 -1.0000e+00 1.0000e+00] * [ x y 1 ]' = 0.
    print( l2 ) # -> l2: ( x, y ) | [1.5000e+00 -1.5000e+00 -1.5000e+00] * [ x y 1 ]' = 0.
    print( l3 ) # -> l3: ( x, y ) | [-1.0000e+00 -1.0000e+00 1.0000e+00] * [ x y 1 ]' = 0.
    print( l4 ) # -> l4: ( x, y ) | [2.0000e+00 2.0000e+00 -2.0000e+00] * [ x y 1 ]' = 0.
    print()

    # The relationships between two lines lx and ly can be:
    # 1) coincident lines: lx == ly or lx.are_coincident( ly ).
    #    Notice that l4 = -2 * l3. From the projective geometry both lines are
    #    coincident because they satisfy the same equation. 
    print( f'Is l1 == l1? {l1 == l1}.' )
    print( f'Is l3 == l4? {l3 == l4}.' )
    print( f'Is l1 == l2? {l1 == l2}.\n' )

    # 2) parallel lines: lx // ly or lx.are_parallel( ly ).
    #    Notice that coincident lines are parallel ones, as well.
    print( f'Is l1 // l2? {l1 // l2}.' )
    print( f'Is l2 // l3? {l2 // l3}.' )
    print( f'Is l3 // l4? {l3 // l4}.\n' )

    # 3) concurrent lines: lx.are_concurrent( ly )
    print( f'Are l1 and l2 concurrent lines? {l1.are_concurrent( l2 )}.' )
    print( f'Are l2 and l3 concurrent lines? {l2.are_concurrent( l3 )}.' )
    print( f'Are l3 and l4 concurrent lines? {l3.are_concurrent( l4 )}.\n' )

    # 4) perpendicular lines: lx + ly or lx.are_perpendicular( ly ).
    print( f'Is l1 + l3? {l1 + l3}.' )
    print( f'Is l2 + l4? {l2 + l4}.' )
    print( f'Is l3 + l4? {l3 + l4}.\n' )

    # Euclidean distance between lines.
    # Notice that, from the theory the distance between two concurrent
    # lines is equal to zero. 
    print( f'Distance from l1 to l2 is {l1.distance( l2 ):.4f}.' )
    print( f'Distance from l2 to l3 is {l2.distance( l3 ):.4f}.' )
    print( f'Distance from l3 to l4 is {l3.distance( l4 ):.4f}.\n' )

    # Lines at infinity.
    l5 = CLine( ( 0, 0, 2 ), 'l5' )
    l6 = CLine( ( 1, -1, 1 ), 'l6' )
    print( l5 )
    print( l6 )
    print( f'Is l5 a line at infinity? {l5.at_infinity()}.' )
    print( f'Is l6 a line at infinity? {l6.at_infinity()}.\n' )

    # Distance from l5 at infinity and l6.
    d56 = l5.distance( l6 )
    print( f'Distance from {l5} to {l6} is {d56}.\n' )

Working with points and lines (v0.2.6 onwards)

Now that I have introduced you to the concepts of points and lines in projective geometry and how you should work with them. The next step is to know how to use both geometric shapes together.

However, before we start this step I must answer a question that you may be wondering. If points and lines have the same vector representation, i.e., they are both vectors in $\mathcal{R}^3$, how will I know if I am using a point or a line?

A generic answer would be: The context will tell you whether the vectors in $\mathcal{R}^3$ are points or lines. For instance, the cross product operation can be used with the following operands:

1. two points: The result is a straight line $l$ that passes through both points.
2. two lines: The result is the point of intersection $p$ between both lines.
3. a point $p$ and a line $l$: The result is a line that is perpendicular to the straight line $l$ and passes through the point $p$.

Finally, the Table below shows six interesting interpretations about vector representation of points and lines, four of which are not well-posed in two-dimensional Euclidean Geometry, namely point and line at infinity.

$$ % \def\arraystretch{2.5} \begin{array}{ccc} \text{vector in }\mathcal{R}^3 & \text{point} & \text{line}\newline \hline\hline [\begin{array}{ccc} \alpha & \beta & 0.0 \end{array}]^{T} & \text{point at infinity} & \text{line passing through the origin}\newline \hline [\begin{array}{ccc} 0.0 & 0.0 & \gamma \end{array}]^{T} & \text{point at origin with }\gamma=1.0 & \text{line at infinity}\newline \hline [\begin{array}{ccc} 0.0 & 0.0 & 0.0 \end{array}]^{T} & \text{point at infinity} & \text{line at infinity}\newline \hline \end{array} $$

• How to work with points and lines in pyConics.

    from pyConics import CPoint, CLine

    # Points.
    p1 = CPoint( ( 1, 3 ), 'p1' )   # p1 = ( 1, 3 )
    p2 = CPoint( ( 0, 4 ), 'p2' )   # p2 = ( 0, 4 )
    p3 = CPoint( ( -2, 0 ), 'p3' )  # p3 = ( -2, 0 )
    p4 = CPoint( ( 2, 0 ), 'p4' )   # p4 = ( 2, 0 )
    p5 = CPoint( ( 0, -4 ), 'p5' )  # p5 = ( 0, -4 )

    # Lines.
    l1 = CLine( ( 2, -1, 4 ), 'l1' )   #l1: y = 2x + 4
    l2 = CLine( ( 2, -1, -4 ), 'l2' )  #l2: y = 2x - 4
    l3 = CLine( ( 2, 1, 0 ), 'l3' )    #l3: y = -2x
    l4 = CLine( ( 1, 0, -2 ), 'l4' )   #l4: x = 2
    l5 = CLine( ( 0, 1, -4 ), 'l5' )   #l5: y = 4
    l6 = CLine( ( 3, -1, 0 ), 'l6' )   #l6: y = 3x

    # Test whether a point is in or is not in a Line.
    print( l1, p1, p2, p3, sep='\n' )
    print( f'Is p1 in l1? {p1 in l1}.' )    # p1 belongs to l1: False
    print( f'Is p2 in l1? {p2 in l1}.' )    # p2 belongs to l1: True
    print( f'Is p3 in l1? {p3 in l1}.\n' )  # p3 belongs to l1: True
    print( l6, p1, p4, sep='\n' )
    print( f'Is p1 in l6? {p1 in l6}.' )    # p1 belongs to l6: True
    print( f'Is p4 in l6? {p4 in l6}.\n' )  # p4 belongs to l6: False
    print( l4, p4, sep='\n' )
    print( f'Is p4 in l4? {p4 in l4}.\n' )  # p4 belongs to l4: True
    print( l5, p2, sep='\n' )
    print( f'Is p2 in l5? {p2 in l5}.\n' )  # p2 belongs to l5: True

    # Use the cross product to handle points and lines.
    # 1) the cross product between two points gives us a line that
    #    passes through these points.
    l: CLine = p2 * p3
    l.name = 'l'
    print( l1, l, sep='\n' )
    print( f'Are l and l1 coincident? {l==l1}.\n' )
    l: CLine = p4 * p5
    l.name = 'l'
    print( l2, l, sep='\n' )
    print( f'Are l and l2 coincident? {l==l2}.\n' )

    # 2) the cross product between two lines gives us their point
    #    of intersection.
    p: CPoint = l1 * l3
    p.name = 'p'
    print( l1, l3, p, sep='\n' )
    print( f'Is p in l1? {p in l1}.' )
    print( f'Is p in l3? {p in l3}.\n' )
    p: CPoint = l2 * l4
    p.name = 'p'
    print( l2, l4, p4, p, sep='\n' )
    print( f'Is p in l2? {p in l2}.' )
    print( f'Is p in l4? {p in l4}.' )
    print( f'Are p and p4 coincident? {p==p4}.\n' )

    # Two parallel lines have their point of intersection at infinity.
    # So, ...
    p: CPoint = l1 * l2
    p.name = 'p'
    print( l1, l2, p, sep='\n' )
    print( f'Is p in l1? {p in l1}.' )
    print( f'Is p in l2? {p in l2}.' )
    print( f'Is p in l4? {p in l4}.' )
    print( f'Is p at infinity? {p.at_infinity()}.\n' )

    # 3) the cross product between a point and a line gives us a
    #    line which is perpendicular to that line and passes
    #    through that point.
    l: CLine = l2 * p2
    l.name = 'l'
    print( p2, l2, l, sep='\n' )
    print( f'Is p2 in l? {p2 in l}.' )
    print( f'Are l2 and l perpendicular? {l2 + l}.\n' )
    p: CPoint = l2 * l
    p.name = 'p'
    print( l2, l, p, sep='\n' )
    print( f'Is p in l? {p in l}.' )
    print( f'Is p in l2? {p in l2}.' )

    # Getting the distance between p2 and l2.
    print( f'Distance from p2 to l2 = {p2.distance( l2 ):.4f}.' )
    print( f'Distance from p2 to p  = {p2.distance( p ):.4f}.\n' )

Creating and setting up the CFigure and CAxes Classes (v1.0.0 onwards)

Two classes have been developed for this version, namely CFigure and CAxes, so that it is now possible to graph points and lines from the CPoint and CLine classes, respectively. Both are derived from pyPlot classes. Therefore, they are very similar to those existing in pyPlot. Actually, you can use the get_pyplot_figure() and get_pyplot_axes() methods to obtain an instance of pyPlot's Figure and Axes Classes, respectively, and work directly with them as if you were using pyPlot.

In this section, code segments will be used for each sub-section, so if you want to get the result shown in our image, the current code segment must be inserted below the previous one.

Let's know how to set up these classes.

• Creating an empty figure and then its four axes.

    from pyConics import CFigure, CAxes

    # Set interactive mode.
    # Activate this mode so that it is not necessary to call the show() method.
    # Whether you comment this line or use CFigure.ioff() method, the show()
    # method must be called.
    CFigure.ion()

    # Create an empty figure.
    # Its width and height are relative to the screen size.
    width = 0.35
    f1: CFigure = CFigure( (width, 16.0 / 9.0 * width ) )

    # If CFigure.ion() is on then you need to press a key to continue.
    if ( CFigure.is_interactive() ):
        input( 'Press any key to continue...' )

    # Create a 2x2 grid of axes from f1.
    f1.create_axes( ( 2, 2 ) )

    # Get the tuple of CAxes classes for the 2x2 grid.
    axes = f1.axes

    # If CFigure.ion() is on then you need to press a key to continue.
    if ( CFigure.is_interactive() ):
        input( 'Press any key to continue...' )

• Changing the x- and y-axis limits and ticks of an axes.

    # Changing the x- and y-axis limits and the
    # x- and y-ticks of axes[ 0 ].
    axes[ 0 ].xlim = ( 0, 2 )
    axes[ 0 ].ylim = ( -1, 1 )
    xtick = np.linspace( 0, 2, 11 )
    ytick = np.linspace( -1, 1, 11 )
    axes[ 0 ].xticks = xtick
    axes[ 0 ].yticks = ytick

    # Changing the x- and y-axis limits and the
    # x- and y-ticks of axes[ 1 ].
    axes[ 1 ].xlim = ( -1, 1 )
    axes[ 1 ].ylim = ( -1, 1 )
    xtick = np.linspace( -1, 1, 11 )
    ytick = np.linspace( -1, 1, 11 )
    axes[ 1 ].xticks = xtick
    axes[ 1 ].yticks = ytick

    # Changing the x- and y-axis limits and the
    # x- and y-ticks of axes[ 2 ].
    axes[ 2 ].xlim = ( -10, 10 )
    axes[ 2 ].ylim = ( -10, 10 )
    xtick = np.linspace( -10, 10, 11 )
    ytick = np.linspace( -10, 10, 11 )
    axes[ 2 ].xticks = xtick
    axes[ 2 ].yticks = ytick

    # Changing the x- and y-axis limits and the
    # x- and y-ticks of axes[ 3 ].
    # Note that the dimension of the plot box has
    # changed. This is because the aspect ration of
    # the CAxes class has equal scaling ( x/y-scaling = 1.0).
    # This makes circles circular, not elliptical.
    axes[ 3 ].xlim = ( -7, 7 )
    axes[ 3 ].ylim = ( -10, 10 )
    xtick = np.linspace( -7, 7, 5 )
    ytick = np.linspace( -10, 10, 11 )
    axes[ 3 ].xticks = xtick
    axes[ 3 ].yticks = ytick

    # If CFigure.ion() is on then you need to press a key to continue.
    if ( CFigure.is_interactive() ):
        input( 'Press any key to continue...' )

• Changing title, x-label, and y-label of an axes.

# Changing the title of axes[ 0 ].
    axes[ 0 ].title = 'The tittle is hello, world!'

    # If CFigure.ion() is on then you need to press a key to continue.
    if ( CFigure.is_interactive() ):
        input( 'Press any key to continue...' )

    # Changing the x- and y-label of axes[ 1 ].
    axes[ 1 ].xlabel = 'this is a physical quantity'
    axes[ 1 ].ylabel = 'this is another physical quantity'

    # If CFigure.ion() is on then you need to press a key to continue.
    if ( CFigure.is_interactive() ):
        input( 'Press any key to continue...' )

    # Using latex language in axes[ 2 ].
    axes[ 2 ].title = f'alpha is written as $\\alpha$'
    axes[ 2 ].xlabel = f'beta is written as $\\beta$'
    axes[ 2 ].ylabel = f'gamma is written as $\\gamma$'

    # If CFigure.ion() is on then you need to press a key to continue.
    if ( CFigure.is_interactive() ):
        input( 'Press any key to continue...' )

NOTE:
The CAxes class in pyConics has setters for its title, xlabel, ylabel, xticks, and yticks that do not allow you to change their font size directly, i.e., the pyConics fixed the font size of these attributes by default, namely:

• title: fontsize = 9
• x- and y-labels: fontsize = 8
• x- and y-ticks: fontsize = 8

If, for any reason, you need to change one of these default font sizes, you must get the Axes object that belongs to the pyPlot by mean of the CAxes's get_pyplot_axes() method and then call the method that does this task.

An example code to do this is shown below.

    from pyConics import CFigure, CAxes

    # Set interactive mode.
    CFigure.ion()

    # Create an empty figure.
    # Its width and height are relative to the screen size.
    width = 0.35
    f1: CFigure = CFigure( (width, 16.0 / 9.0 * width ) )

    # Create a 1x1 grid of axes from f1.
    # The title font size is 9.
    f1.create_axes( ( 1, 1 ) )

    # If CFigure.ion() is on then you need to press a key to continue.
    if ( CFigure.is_interactive() ):
        input( 'Press any key to continue...' )

    # Get the tuple of CAxes classes for the 1x1 grid.
    ax = f1.axes

    # Get the pyPlot's axes.
    pp_ax = ax[ 0 ].get_pyplot_axes()

    # Changing the font size of the title in pyPlot's Axes.
    # Now, the title font size is 16.
    pp_ax.set_title( pp_ax.get_title(), fontsize = 16 ) 

    # If CFigure.ion() is on then you need to press a key to continue.
    if ( CFigure.is_interactive() ):
        input( 'Press any key to continue...' )

Drawing points (CPoint) and lines (CLine) on CAxes Class (v1.0.0 onwards)

Once you have already learned how to create and set up the CFigure and CAxes classes, you are able to learn how to use the plot() method in CAxes class for drawing points and lines inside the CPoint and CLine classes, respectively.

This plot() method has been derived from the pyPlot's plot() method. Therefore, everything you did with pyPlot's plot(), you can continue to do with plot() method that belongs to the CAxes class.

In addition, CAxes.plot() method can receive either a point such as CPoint, a straight line such as Cline, a list of points such as list[CPoint] or a list of straight lines such as list[CLine] as argument to be plotted on screen (into a CAxes class).

Now, let's start.

• Plotting a CPoint and a numpy point.

    from pyConics import CFigure, CAxes
    from pyConics import CPoint
    import numpy as np 

    # Create an empty figure.
    # Its width and height are relative to the screen size.
    width = 0.35
    f: CFigure = CFigure( (width, 16.0 / 9.0 * width ) )

    # Create a 1x1 grid of axes from f.
    # The title font size is 9.
    f.create_axes( ( 1, 1 ) )

    # Get the tuple of CAxes classes for the 1x1 grid.
    ax = f.axes

    # Creating a point using numpy.
    p1 = np.array( [ 0.5, 0.6 ] )       # p1 =( 0.5, 0.6 ) 

    # Creating a point using CPoint.
    p2 = CPoint( ( 0.5, 0.4 ), 'p2' )   # p2 =( 0.5, 0.4 )

    # Plotting both points
    ax[ 0 ].plot( p1[ 0 ], p1[ 1 ], 'ob', p2, 'om' )

    # Show Figure on screen.
    CFigure.show()

• Plotting a list of CPoint and a list of numpy points.

    from pyConics import CFigure, CAxes
    from pyConics import CPoint
    import numpy as np 

    # Create an empty figure.
    # Its width and height are relative to the screen size.
    width = 0.35
    f: CFigure = CFigure( (width, 16.0 / 9.0 * width ) )

    # Create a 1x1 grid of axes from f.
    # The title font size is 9.
    f.create_axes( ( 1, 1 ) )

    # Get the tuple of CAxes classes for the 1x1 grid.
    ax = f.axes

    # Creating a list of points using numpy.
    p = np.array( [ [ 0.3, 0.1 ],       # p[ 0 ] = ( 0.3, 0.1 )
                    [ 0.3, 0.2 ],       # p[ 1 ] = ( 0.3, 0.2 )
                    [ 0.3, 0.3 ],       # p[ 2 ] = ( 0.3, 0.3 )
                    [ 0.3, 0.4 ],       # p[ 3 ] = ( 0.3, 0.4 )
                    [ 0.3, 0.5 ] ] )    # p[ 4 ] = ( 0.3, 0.5 )

    # Creating a list of points using CPoint.
    p1 = CPoint( ( 0.7, 0.5 ), 'p1' )   # p1 = ( 0.7, 0.5 )
    p2 = CPoint( ( 0.7, 0.6 ), 'p2' )   # p2 = ( 0.7, 0.6 )
    p3 = CPoint( ( 0.7, 0.7 ), 'p3' )   # p3 = ( 0.7, 0.7 )
    p4 = CPoint( ( 0.7, 0.8 ), 'p4' )   # p4 = ( 0.7, 0.8 )
    p5 = CPoint( ( 0.7, 0.9 ), 'p5' )   # p5 = ( 0.7, 0.9 )
    pl = [ p1, p2, p3, p4, p5 ]

    # Plotting both the list of points
    ax[ 0 ].plot( p[ :, 0 ], p[ :, 1 ], 'ob', pl, '^m', markersize = 8 )

    # Show Figure on screen.
    CFigure.show()

• Plotting a CLine and a numpy line.

A CLine is drawn with a number of points determined by the clinesamples parameter, with its default value being 11.

    from pyConics import CFigure, CAxes
    from pyConics import CLine
    import numpy as np 

    # Create an empty figure.
    # Its width and height are relative to the screen size.
    width = 0.35
    f: CFigure = CFigure( (width, 16.0 / 9.0 * width ) )

    # Create a 1x1 grid of axes from f.
    # The title font size is 9.
    f.create_axes( ( 1, 1 ) )

    # Get the tuple of CAxes classes for the 1x1 grid.
    ax = f.axes

    # Creating a line using numpy.
    x_i, x_f = ax[ 0 ].xlim
    x = np.linspace( x_i, x_f, 11 )
    y = 1.05 * x                            #  y = 1.05x

    # Creating a line using CLine.
    l1 = CLine( ( 1.05, 1.0, -1.0 ), 'l1' ) # y = -1.05x + 1.0 

    # Plotting both lines ( clinesamples = 11 )
    ax[ 0 ].plot( x, y, 'ob-', l1, 'sm-', linewidth = 0.5, markersize = 6 )

    # Plotting a vertical line with clinesamples = 21
    l2 = CLine( ( 1.0, 0.0, -0.8 ), 'l2' )  # x = 0.8 for all y.
    ax[ 0 ].plot( l2, '^y-', clinesamples = 21, linewidth = 0.5, markersize = 6 )

    # Show Figure on screen.
    CFigure.show()

• Plotting a list of CLine.

    from pyConics import CFigure, CAxes
    from pyConics import CLine
    import numpy as np 

    # Create an empty figure.
    # Its width and height are relative to the screen size.
    width = 0.35
    f: CFigure = CFigure( (width, 16.0 / 9.0 * width ) )

    # Create a 1x1 grid of axes from f.
    # The title font size is 9.
    f.create_axes( ( 1, 1 ) )

    # Get the tuple of CAxes classes for the 1x1 grid.
    ax = f.axes

    # Create some lines and add them to a list.
    l1 = CLine(( 0.0, 1.0, -0.1 ), 'l1' )   # y = 0.1 for all x
    l2 = CLine(( 0.0, 1.0, -0.9 ), 'l2' )   # y = 0.9 for all x
    l3 = CLine(( 1.0, 0.0, -0.1 ), 'l3' )   # x = 0.1 for all y
    l4 = CLine(( 1.0, 0.0, -0.9 ), 'l4' )   # x = 0.9 for all y
    l = [ l1, l2, l3, l4 ]
    ax[ 0 ].plot( l, 'ob-', clinesamples = 11, linewidth = 0.5, markersize = 4 )

    # Create some others lines and add them to a list.
    l5 = CLine(( 1.0, 1.0, -1.0 ), 'l5' )   # y = -x + 1
    l6 = CLine(( 1.0, -1.0, 0.0 ), 'l6' )   # y = x
    l = [ l5, l6 ]
    ax[ 0 ].plot( l, 'or:', clinesamples = 21, linewidth = 1.0, markersize = 6 )

    # Show Figure on screen.
    CFigure.show()

• Plotting line segments through a sequence of points.

When you are using the plot() method to graph a line, the result on screen is a straight line that crosses the entire range of the $x$-axis and $y$-axis.

However, if you only want to get a few points from this line (line segment), you might want to use the sequence() method of the CLine class.

For instance, given a CLine $l$ and a sequence $x = [ x_1, x_2, \cdots, x_{n-1}, x_n ]$, the sequence() method retuns a list of CPoint $p_s$ such that,

$$ p_s = \lbrack \begin{array}{ccccc} \text{CPoint}( ( y_1, x_1 ), & \text{CPoint}( ( y_2, x_2 ) ), & \cdots, & \text{CPoint}( ( y_{n-1}, x_{n-1} ), & \text{CPoint}( ( y_{n}, x_{n} ) ) \end{array} \rbrack $$

    from pyConics import CFigure, CAxes
    from pyConics import CPoint, CLine
    import numpy as np 

    # Create an empty figure.
    # Its width and height are relative to the screen size.
    width = 0.35
    f: CFigure = CFigure( (width, 16.0 / 9.0 * width ) )

    # Create a 1x1 grid of axes from f.
    # The title font size is 9.
    f.create_axes( ( 1, 1 ) )

    # Get the tuple of CAxes classes for the 1x1 grid.
    ax = f.axes

    # Create some lines.
    l1 = CLine(( 0.0, 1.0, -0.1 ), 'l1' )   # y = 0.1 for all x
    l2 = CLine(( 1.0, 0.0, -0.1 ), 'l2' )   # x = 0.1 for all y
    l3 = CLine(( 1.0, -1.0, 0.0 ), 'l3' )   # y = x

    # Plot that lines.
    ax[ 0 ].plot( l1, 'b-', linewidth = 1.5 )
    ax[ 0 ].plot( l2, 'b-', linewidth = 1.5 )
    ax[ 0 ].plot( l3, 'b-', linewidth = 1.5 )

    # Create a range from 0.3 to 0.8 with step = 0.5.
    x = np.linspace( 0.3, 0.8, 11 )

    # Get a sequence of points form these lines.
    ps1 = list( l1.sequence( list( x ) ) )
    ps2 = list( l2.sequence( list( x ) ) )
    ps3 = list( l3.sequence( list( x ) ) )

    # Plot these line segments as a sequence of points.
    ax[ 0 ].plot( ps1, 'ro', markersize = 6 )
    ax[ 0 ].plot( ps2, 'ro', markersize = 6 )
    ax[ 0 ].plot( ps3, 'ro', markersize = 6 )

    # Show Figure on screen.
    CFigure.show()

WARNING:
It is worth mentioning that, when you have a line $l$ that is parallel to the $y$-axis, the input sequence $x$ will be assigned to a variable $y$ that will sweep the $y$-axis instead of the $x$-axis. For a better understanding, pay attention to the variable l2 in the code segment above.

• Solving a simple problem of geometry with points and lines.

Finally, this is the last subsection about how to draw points (CPoint) and lines (CLine) on CAxes class. Here, you are going to learn how to solve a simple problem of geometry using pyConics. Are you ready? Let's do it.

Problem formulation:

Given two points $p_1=( -0.4, 0.6 )$ and $p_2=( 0.0, -0.8 )$ and a line $l_1 = ( 0.9, -1.0, 0.1 )$ as shown in the figure below, solve the following geometry problems:

1. Get a line $l_2$ that passes through these points.
2. Get a point $p_3$ that is the point of intersection between the lines $l_1$ and $l_2$.
3. Get a line $l_3$ that passes through the point $p_1$ and is perpendicular to the line $l_1$.

Graph $l_2$, $p_3$, and $l_3$ on the same canvas and also evaluate the results with the in operator and the are_perpendicular() method.

Problem resolution:

    from pyConics import CFigure, CAxes
    from pyConics import CPoint, CLine
    import numpy as np 

    # Create an empty figure.
    # Its width and height are relative to the screen size.
    width = 0.35
    f: CFigure = CFigure( (width, 16.0 / 9.0 * width ) )

    # Create a 1x1 grid of axes from f.
    # The title font size is 9.
    f.create_axes( ( 1, 1 ) )

    # Get the tuple of CAxes classes for the 1x1 grid.
    ax = f.axes

    # Define a title.
    ax[ 0 ].title = 'A Simple Problem of Geometry with Points and Lines'

    # Change its axis.
    ax[ 0 ].xlim = ( -1, 1 )
    ax[ 0 ].xticks = np.linspace( -1, 1, 11 )
    ax[ 0 ].ylim = ( -1, 1 )
    ax[ 0 ].yticks = np.linspace( -1, 1, 11 )

    # Create and plot the geometric forms of the problem.
    p1 = CPoint( ( -0.4, 0.6 ), 'p_1' )
    p2 = CPoint( ( 0.0, -0.8 ), 'p_2' )
    l1 = CLine( ( 0.9, -1.0, 0.1 ), 'l_1' ) # y = 0.9x + 0.1
    ax[ 0 ].plot( p1, 'or', p2, 'or', l1, 'b-', clinesamples = 21 )
    ax[ 0 ].text( p1.x, p1.y, f'${p1.name}$', ha = 'left', va = 'bottom' )
    ax[ 0 ].text( p2.x, p2.y, f'${p2.name}$', ha = 'left', va = 'bottom' )
    ls = l1.sequence( [ 0.6 ] )[ 0 ]
    ax[ 0 ].text( ls.x, ls.y, f'${l1.name}$', ha = 'right', va = 'bottom' )

    # Solving the problem (1):
    l2: CLine = p1 * p2
    l2.name = 'l_2'
    print( l2 )
    ax[ 0 ].plot( l2, 'y-', clinesamples = 21 )
    ls = l2.sequence( [ -0.1 ] )[ 0 ]
    ax[ 0 ].text( ls.x, ls.y, f'${l2.name}$', ha = 'left', va = 'bottom' )
    print( 'Evaluating the result for the problem (1):' )
    print( f'Does p1 lie in l2? {p1 in l2}' ) # True
    print( f'Does p2 lie in l2? {p2 in l2}' ) # True
    print()

    # Solving the problem (2):
    p3: CPoint = l1 * l2
    p3.name = 'p_3'
    print( p3 )
    ax[ 0 ].plot( p3, 'om' )
    ax[ 0 ].text( p3.x, p3.y, f'${p3.name}$', ha = 'right', va = 'bottom' )
    print( 'Evaluating the result for the problem (2):' )
    print( f'Does p3 lie in l1? {p3 in l1}' ) # True
    print( f'Does p3 lie in l2? {p3 in l2}' ) # True
    print()

    # Solving the problem (3):
    l3: CLine = p1 * l1
    l3.name = 'l_3'
    print( l3 )
    ax[ 0 ].plot( l3, 'g-' )
    ls = l3.sequence( [ 0.4 ] )[ 0 ]
    ax[ 0 ].text( ls.x, ls.y, f'${l3.name}$', ha = 'left', va = 'bottom' )
    print( 'Evaluating the result for the problem (3):' )
    print( f'Does p1 lie in l3? {p1 in l3}' )           # True
    print( f'Are l1 and l3 perpendicular? {l1 + l3}' )  # True
    print()

    # Show Figure on screen.
    CFigure.show()

Contributing

Interested in contributing? Check out the contributing guidelines. Please note that this project is released with a Code of Conduct. By contributing to this project, you agree to abide by its terms.

License

pyConics was created by Jefferson Osowsky. It is licensed under the terms of the GNU General Public License v3.0 license.

Credits

pyConics was created with cookiecutter and the py-pkgs-cookiecutter template.

pyConics has used the following Python packages since its creation:

• numpy
• matplotlib (pyPlot)
• PyAutoGUI
• pytest