File:Alhazen hyperbola.svg
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Original file (SVG file, nominally 400 × 400 pixels, file size: 3 KB)
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Summary
| Description |
English: A geometric solution to Alhazen's problem, seeking the point (red) at which two given points (blue) reflect to each other on a mirrored circle (black). The solution involves the inversions of the two given points through the circle (yellow) and the midpoint of the two inversion points (also yellow). A right hyperbola (red) passes through the two inversion points, centered at the midpoint of these two points, with its asymptotic lines (yellow, through midpoint) parallel and perpendicular to the bisector of the angle subtended by the given points at the circle center. The desired reflection point (red) is one of the crossing points of this hyperbola with the given circle. |
| Date | |
| Source | Own work |
| Author | David Eppstein |
Source code
from PADS.SVG import *
import sys
from cmath import phase
from math import pi
scale=100
bbox = (4+4j)*scale
origin=(1.5+2.5j)*scale
pointrad=4
smallrad=3
svg = SVG(bbox,sys.stdout)
def place(p):
return p*scale+origin
a = -0.25-1.1j
b = 2.2-1.1j
ainv = 1/a.conjugate()
binv = 1/b.conjugate()
invmid = (ainv+binv)/2
bisector = a/abs(a)+b/abs(b)
bisector /= abs(bisector)
def angle(p,q,r):
"""Clockwise turning angle"""
pq = (p-q)/abs(p-q)
qr = (q-r)/abs(q-r)
return pi+phase(pq/qr)
left = a/abs(a)
right = b/abs(b)
for i in range(20):
reflex = (left+right)/abs(left+right)
if angle(a,reflex,0) > angle(0,reflex,b):
left = reflex
else:
right = reflex
svg.circle(origin,scale,fill="none",stroke=colors.black) # unit circle
svg.polyline([place(a),place(reflex),place(b)],fill="none",stroke=colors.blue)
hyperx = [1.1,1.3,1.6,2,2.5,3.1,3.7,4.4,5.2,6.1,7.1,8.2,9.4,11,13]
hyperx = [1/x for x in reversed(hyperx)]+hyperx
unscaled = (ainv-invmid)/bisector
hyperscale = (unscaled.real*unscaled.imag)**0.5*bisector
hyperpos = [(x + 1j/x)*hyperscale+invmid for x in hyperx]
hyperneg = [(-x - 1j/x)*hyperscale+invmid for x in hyperx]
thin={"stroke-width":"0.5"}
svg.group(fill="none",style=thin,stroke=colors.yellow)
svg.segment(place(0),place(a*4))
svg.segment(place(0),place(b*2))
svg.segment(place(-4*bisector),place(4*bisector))
svg.segment(place(-4j*bisector),place(4j*bisector))
svg.segment(place(-4*bisector+invmid),place(4*bisector+invmid))
svg.segment(place(-4j*bisector+invmid),place(4j*bisector+invmid))
svg.ungroup()
svg.group(fill="none",stroke=colors.red)
svg.polycurve([place(q) for q in hyperpos])
svg.polycurve([place(q) for q in hyperneg])
svg.ungroup()
svg.group(style=thin)
svg.group(fill=colors.blue,stroke=colors.black)
svg.circle(place(a),pointrad)
svg.circle(place(b),pointrad)
svg.ungroup()
svg.group(fill=colors.yellow,stroke=colors.black)
svg.circle(place(ainv),smallrad)
svg.circle(place(binv),smallrad)
svg.circle(place(invmid),smallrad)
svg.ungroup()
svg.circle(place(reflex),smallrad,fill=colors.red,stroke=colors.black)
svg.ungroup()
svg.close()
Licensing
I, the copyright holder of this work, hereby publish it under the following license:
 
|
This file is made available under the Creative Commons CC0 1.0 Universal Public Domain Dedication. |
| The person who associated a work with this deed has dedicated the work to the public domain by waiving all of their rights to the work worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law. You can copy, modify, distribute and perform the work, even for commercial purposes, all without asking permission.
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File history
Click on a date/time to view the file as it appeared at that time.
| Date/Time | Thumbnail | Dimensions | User | Comment | |
|---|---|---|---|---|---|
| current | 01:15, 8 June 2025 | | 400 × 400 (3 KB) | David Eppstein | Uploaded own work with UploadWizard |
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