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Constraints on conical coffee cup constructions of cardioids & catacaustics
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The Mathologer video Times Tables, Mandelbrot and the Heart of Mathematics discusses several relationships. For the n=2 and 3 cases, the cardiod and catacaustic (or nephroid per @Rahul's comment) curves are shown in reflections in coffee cups.
In the case of the cardiod the cup must be conical but presumably not cylindrical, for the catacaustic it must be a cylinder.
If I wanted to construct these caustics using Blender, what are the specific constraints on the shapes of the coffee cups and directions of illumination? Can the former be any converging or diverging cone, and does the illumination direction need to have the same angle as that of the cone? Must the later be a perfect cylinder and the illumination only oblique?
"Bonus points" for a description of any possible cup and illumination configuration that could work for n=4.
geometry geometric-construction
asked Jul 23 at 7:30
uhoh
421415
add a comment |Â
up vote
5
down vote
favorite
The Mathologer video Times Tables, Mandelbrot and the Heart of Mathematics discusses several relationships. For the n=2 and 3 cases, the cardiod and catacaustic (or nephroid per @Rahul's comment) curves are shown in reflections in coffee cups.
In the case of the cardiod the cup must be conical but presumably not cylindrical, for the catacaustic it must be a cylinder.
If I wanted to construct these caustics using Blender, what are the specific constraints on the shapes of the coffee cups and directions of illumination? Can the former be any converging or diverging cone, and does the illumination direction need to have the same angle as that of the cone? Must the later be a perfect cylinder and the illumination only oblique?
"Bonus points" for a description of any possible cup and illumination configuration that could work for n=4.
geometry geometric-construction
asked Jul 23 at 7:30
uhoh
421415
3
Cunning combination of classifying concepts in the caption!
– joriki
Jul 23 at 7:48
1
"Catacaustic" is a general term encompassing both the cardioid and the curve in the second image. The latter is, specifically, a nephroid.
– Rahul
Jul 23 at 9:40
add a comment |Â
up vote
5
down vote
favorite
up vote
5
down vote
favorite
The Mathologer video Times Tables, Mandelbrot and the Heart of Mathematics discusses several relationships. For the n=2 and 3 cases, the cardiod and catacaustic (or nephroid per @Rahul's comment) curves are shown in reflections in coffee cups.
In the case of the cardiod the cup must be conical but presumably not cylindrical, for the catacaustic it must be a cylinder.
If I wanted to construct these caustics using Blender, what are the specific constraints on the shapes of the coffee cups and directions of illumination? Can the former be any converging or diverging cone, and does the illumination direction need to have the same angle as that of the cone? Must the later be a perfect cylinder and the illumination only oblique?
"Bonus points" for a description of any possible cup and illumination configuration that could work for n=4.
geometry geometric-construction
asked Jul 23 at 7:30
uhoh
421415
The Mathologer video Times Tables, Mandelbrot and the Heart of Mathematics discusses several relationships. For the n=2 and 3 cases, the cardiod and catacaustic (or nephroid per @Rahul's comment) curves are shown in reflections in coffee cups.
In the case of the cardiod the cup must be conical but presumably not cylindrical, for the catacaustic it must be a cylinder.
If I wanted to construct these caustics using Blender, what are the specific constraints on the shapes of the coffee cups and directions of illumination? Can the former be any converging or diverging cone, and does the illumination direction need to have the same angle as that of the cone? Must the later be a perfect cylinder and the illumination only oblique?
"Bonus points" for a description of any possible cup and illumination configuration that could work for n=4.
geometry geometric-construction
asked Jul 23 at 7:30
uhoh
421415
edited Jul 24 at 2:42
asked Jul 23 at 7:30
uhoh
421415
asked Jul 23 at 7:30
uhoh
421415
asked Jul 23 at 7:30
uhoh
421415
421415
3
Cunning combination of classifying concepts in the caption!
– joriki
Jul 23 at 7:48
1
"Catacaustic" is a general term encompassing both the cardioid and the curve in the second image. The latter is, specifically, a nephroid.
– Rahul
Jul 23 at 9:40
add a comment |Â
3
Cunning combination of classifying concepts in the caption!
– joriki
Jul 23 at 7:48
1
"Catacaustic" is a general term encompassing both the cardioid and the curve in the second image. The latter is, specifically, a nephroid.
– Rahul
Jul 23 at 9:40
3
3
Cunning combination of classifying concepts in the caption!
– joriki
Jul 23 at 7:48
Cunning combination of classifying concepts in the caption!
– joriki
Jul 23 at 7:48
1
1
"Catacaustic" is a general term encompassing both the cardioid and the curve in the second image. The latter is, specifically, a nephroid.
– Rahul
Jul 23 at 9:40
"Catacaustic" is a general term encompassing both the cardioid and the curve in the second image. The latter is, specifically, a nephroid.
– Rahul
Jul 23 at 9:40
add a comment |Â
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3
Cunning combination of classifying concepts in the caption!
– joriki
Jul 23 at 7:48
1
"Catacaustic" is a general term encompassing both the cardioid and the curve in the second image. The latter is, specifically, a nephroid.
– Rahul
Jul 23 at 9:40