Euclid's Muse
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Reflections
A collection of examples which explore the properties and uses of reflections and combinations of reflections.Tags: reflection, transformation
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Collection Contents:
Reflection Composition
Composition of reflection in two parallel lines. Can you match the combined reflections with a translation?
Reflection Composition 2
Composition of reflections in two non-parallel lines. can you match a rotation to the combined reflection?
Reflection Clock
Press the Reflect button to reflect the triangle in the minute hand. Press again to reflect the result in the second hand. Can you describe the motion of the second reflected image?
Kaleidoscope
A kaleidoscope works by mirrors. Specifically an equilateral triangle of mirrors which present reflections and reflections of reflections. Here is the inner core. You can drag the red point to rotate the kaleidoscope. You can also drag the vertices of the original triangles.
Parabolic Reflector
A parabola reflects rays which are parallel to its axis of symmetry through its focus. What happens when the rays are not parallel to the axis?
Locus of intersection of reflected rays
The reflection of parallel rays from symmetric points on a parabola intersect on a circle
Parabolic Solar Cooker
Explore the relationship between f-number of a parabolic solar cooker and its sensitivity to change in angle of incoming light
Tchirnhausen's Cubic
The caustic formed by light projecting perpendicular to the axis of a parabola is called Tchirnhausen's Cubic. What happens when the light projects at some other angle?
Minimum Inscribed Perimeter
Fagnano's problem Find the inscribed triangle with minimum perimeter. You can experiment, or press the Reflect button to see how reflections can help with this problem.
Minimum Inscribed Perimeter 2
The problem – find the triangle with minimal perimeter that can be inscribed in a given triangle. The solution: reflection helps reduce the problem to a single parameter. The answer: is geometrically appealing. Can you solve it without using the hint?
Pascal's Limacon
Given a circle with center A and a point C, here are three related ways of constructing Pascal’s Limacon: It is the isotomic (look it up) of the circle with pole C. It is the conchoid (look it up) of a circle centered at A whose circumference passes through C. It is an epitrochoid (look it up) formed by a circle of equal radius rolling around the original. This app constitutes a visual proof of the above, and depends on the fact that the composition of reflections in two parallel lines is equivalent to a translation of twice the distance between the lines.
Box Solar Cooker
A simple box solar cooker works by reflecting sunlight from its lid into the box. Can you work out a relationship between the angle of the sunlight and the best angle to open the lid? Can you prove it?
Optimal Performance of Box Solar Cooker
The solar concentration ratio of a solar cooker is the ratio of the amount of sunlight concentrated on the target to the size of the target. This model presents the best situation for a given lid angle, and lets you see how much light can be captured at that lid angle. The question is: what lid angle captures the most sunlight?
Oil Pipeline Problem
Where is the best place to put a pumping station to connect two refineries to an existing oil pipeline? This is a classic problem involving finding teh shortest distance between two points but
Ellipse Reflection Property
One definition of an ellipse is as the locus of points the sum of whose distance from two fixed points (the foci) is constant. This gives you a way to draw the ellipse with two pins and a piece of string. Here we relate that definition to the focal property: that light emanating from one focus reflects to the other focus.
Coffee Cup Caustic
Light rays emanate from point C and reflects in the edge of a circle (the coffee cup). The caustic is the phantom curve which appears where the rays concentrate.
Caustic Formation
Light from C reflects in the tangent to the circle at point B. Observe the trace as B rotates around the circle. While the individual rays move, the phantom curve - the caustic - does not. The caustic is the mathematical envelope of the reflected rays, which is to say that each reflected ray is tangential to the envelope curve. At any moment, each ray is moving with a combination of rotation and translation. Each point on the ray is undergoing a translation. Points on the ray which lie on the caustic are moving along teh length of the ray, and hence their motion is not visible. Which is why the caustic looks stationary while everything else moves.
Caustic Cusps
The caustic curve is shown for light emanating from point D. Can you relate the location of the caustic’s cusps to the position of D? Concentric circles are in increments of 0.1 radius
Caustic Cusps
Find the direction of the rays which cause the cusps. What do you notice about the reflection angle here?
Maximum Angle of Reflection
Drag B to find the maximum angle of reflection. The circumcircle of ABC is the locus of all points which subtend the same angle on chord AC. In general G is inside the circumcircle and so angle AGC < angle ABC. The exception is when B=G.