Fitting an ellipse into an isoceles triangle

tk_iso_ellipse0.py

#! /usr/bin/env python3

''' iso_ellipse. An ellipse that touches an isoceles triangle at the 
    midpoint of its base and at a given distance on the sides.

    The apex of the triangle is at the origin, with the triangle's axis
    on the +X axis. The slopes of the equal sides are m and -m,
    the "height" is V, so the other 2 vertices are at (V, mV) and (V, -mV).

    We want an ellipse that's tangential to the triangle at (X, Y) and (X, -Y)
    where Y = mX. We also want the right vertex of the ellipse at (V, 0)

    Let the parametric equations of the ellipse be
    x = c + a cos t
    y = b sin t

    The right vertex is at t = 0, so V = c + a, i.e.,
    a = V - c

    For all t, dy/dx = (b cos t) / (-a sin t)

    Let t = T at the point of tangency.
    When t = T, y / x = dy / dx = Y / X = m
    Thus y * dx = x * dy
    (b sin T)(-a sin T) = (c + a cos T)(b cos T)
    -ab sin^2 T = bc cos T + ab cos^2 T
    -bc cos T = ab(sin^2 T + cos^2 T)
    b == 0 is a degenerate solution: a horizontal line on the X axis.
    So assuming b > 0,
    -c cos T = a
    cos T = -a / c

    sin^2 T = 1 - cos^2
    = (c^2 - a^2) / c^2
    Let g^2 = c^2 - a^2. Then
    sin T = g / c

    m = dy / dx = (b cos T) / (-a sin T)
    = (b * -a / c) / (-a * g / c)
    m = b / g
    b = mg

    X = c + a cos T = c - a^2 / c
    = (c^2 - a^2) / c
    X = g^2 / c

    Y = b sin T = bg / c = mX

    c^2 = a^2 + g^2
    c^2 = (V - c)^2 + Xc
    c^2 = V^2 + c^2 -2Vc + Xc
    (2V - X)c = V^2
    c = V^2 / (2V - X)
    c = V / (2 - X / V)

    Written by PM 2Ring 2017.09.25
'''

import tkinter as tk
from math import sqrt

class Gui:
    def __init__(self):
        self.root = root = tk.Tk()
        w, h = root.winfo_screenwidth(), root.winfo_screenheight()
        root.geometry(f'{w}x{h}')
        root.title('Iso Ellipse')

        self.canvas = canvas = tk.Canvas(root)
        canvas.pack(fill='both', expand=True, anchor='n')
        canvas.bind("<Button-1>", self.on_button)

        # A Scale to control the point of tangency
        res = 0.001
        self.scale = tk.Scale(root, from_=res, to=1 - res, resolution=res,
            relief=tk.SUNKEN, orient=tk.HORIZONTAL, command=self.on_scale)
        self.scale.pack(fill='x', anchor='s')
        self.scale_value = 0.5
        self.scale.set(self.scale_value)

        # Draw axes
        self.ox, self.oy = 0, h // 2
        canvas.create_line((0, self.oy, w, self.oy))
        canvas.create_line((self.ox, 20, self.ox, h - 20))
        self.wx = self.wy = self.triangle = self.ellipse = None

        root.mainloop()

    def on_scale(self, value):
        value = float(value)
        if value == self.scale_value:
            return
        self.scale_value = value
        self.draw()

    def on_button(self, event):
        self.wx, self.wy = event.x, event.y
        self.draw()

    def draw_ellipse(self, x, y, v):
        ox, oy = self.ox, self.oy
        canvas = self.canvas
        if self.ellipse:
            canvas.delete(self.ellipse)

        # Compute ellipse parameters. See above for details
        c = v / (2 - x / v)
        a = v - c
        b = sqrt(x * c) * y / x
        bbox = ox + c - a, oy - b, ox + c + a, oy + b
        self.ellipse = canvas.create_oval(bbox, outline='blue', fill='') 

    def draw_triangle(self, v, h):
        ox, oy = self.ox, self.oy
        canvas = self.canvas
        if self.triangle:
            canvas.delete(self.triangle)
        points = ox, oy, ox + v, oy + h, ox + v, oy - h
        self.triangle = canvas.create_polygon(points, outline='red', fill='')

    def draw(self):
        if self.wx is None:
            return
        v = self.wx - self.ox
        if v <= 0:
            return
        h = abs(self.wy - self.oy)
        self.draw_triangle(v, h)

        r = self.scale_value
        self.draw_ellipse(r * v, r * h, v)

Gui()

Click anywhere to draw a triangle, then use the slider to select the point where the ellipse touches the sides of the triangle.