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# Topic: Nephroid

###### In the News (Fri 19 Apr 19)

 nephroid In mathematical terms, this means that the nephroid is the catacaustic (see caustic curve) of a circle when the light source is at infinity, a fact first demonstrated by Christiaan Huygens in 1678 and published by him in his Traité de la lumière (1690). Specifically, the nephroid is the epicycloid formed by a circle of radius a rolling around the outside on a fixed circle of radius 2a. The nephroid is the involute of Cayley's sextic and is also the envelope of circles with their centers on a given circle, touching a given diameter of that circle. www.daviddarling.info /encyclopedia/N/nephroid.html   (394 words)

 Xah: Special Plane Curves: Nephroid Nephroid is the catacaustic of a circle with parallel rays. Nephroid is the envelope of a set of circles with centers on a base circle B and tangent to a fixed diameter of B. Proof. Base circle are not shown) Nephroid is the envelope of a diameter of the circle that generates a cardioid as an epicycloid. xahlee.org /SpecialPlaneCurves_dir/Nephroid_dir/nephroid.html   (699 words)

 Nephroid Definition / Nephroid Research   (Site not responding. Last check: 2007-10-15) The nephroid is an epicycloidIn geometry, an epicycloid is a plane curve produced by tracing the path of a chosen point of a circle — named epicycle — which rolls around without slipping around a fixed circle. nephroid is the catacaustic of a cardioid for a luminous cusp. nephroid is the catacaustic for rays originating at the cusp of a cardioid and reflected by it. www.elresearch.com /Nephroid   (122 words)

 Nephroid The nephroid is the epicycloid formed by a circle of radius a rolling externally on a fixed circle of radius 2a. Huygens, in 1678, showed that the nephroid is the catacaustic of a circle when the light source is at infinity. The involute of the nephroid is Cayley's sextic or another nephroid since they are parallel curves. www-groups.dcs.st-and.ac.uk /~history/Curves/Nephroid.html   (189 words)

 [No title] Special cases are: cardioid (b=a), nephroid (b=a/2)." epicycloidunitspeed::usage="s->epicycloidunitspeed[a,b][s] is a unit-speed parametrization of an epicycloid." epitrochoid::usage="t->epitrochoid[a,b,h][t] is the parametrized curve that is traced out by a point P attached to a circle (of radius b) rolling outside another circle (of radius a). The polar equation is r=a*E^(b*theta)." nephroid::usage="t->nephroid[a][t] is a nephroid or kidney shaped curve. A nephroid is a special case of an epicycloid." nielsenspiral::usage="t->nielsenspiral[a][t] is Nielsen's spiral of radius a." parabola::usage="t->parabola[a][t] is the vertical parabola with vertex at the origin, focus at {0,a} and directrix t->{t,-a}. xtsunxet.usc.es /cordefg/curves2d.m   (1099 words)

 Nephroid - a special epicycloid   (Site not responding. Last check: 2007-10-15) Nephroids can be defined as the trace of a point fixed on a circle of radius 1/2 r that rolls around a fixed circle with radius r. Another way to say it is the Nephroid is a trace of a fixed point on a circle rolling around a fixed circle that has a radius two times as large as the moving circle's. Nephroid, which means kidney-shape, was used by Proctor to describe a two cusped epicycloid. online.redwoods.cc.ca.us /instruct/darnold/calcproj/fall97/dwhite/nephroid.html   (272 words)

 Freeth's Nephroid   (Site not responding. Last check: 2007-10-15) Freeth's Nephroid is a team of great minds who gather weekly in a pub called the Beehive, somewhere in the sticks of Surrey. This rocked the older members of Freeth's Nephroid to the core, but such is their dedication that they still go and sit around the little wooden tables in their high-backed chairs next to the tacky bee poster on the north wall and quietly grumble about the good old days. To be compiled by the Freeth's Nephroid archivist. www.isg.rhul.ac.uk /~martin/freeth/freeth.htm   (625 words)

 The Nephroid Lab: Introduction   (Site not responding. Last check: 2007-10-15) The apparent curve in the first image is the pattern formed on the bottom of an empty flat-bottomed coffee mug when parallel light rays come in and bounce off the circular sides. The similar looking curve in the second image is the path traced out by a point on the boundary of a small disk as the disk rolls along the boundary of a larger disk having twice the diameter. How to find a parametrization in terms of trigonometric functions for curves (like the nephroid) that may be generated by rolling one circle along the outside of another circle. www.geom.uiuc.edu /~fjw/calc-init/nephroid   (306 words)

 nephroid   (Site not responding. Last check: 2007-10-15) the nephroid is the evolute of Cayley's sextic the nephroid is the envelope of a set of circles with centers on a circle and tangent to a diameter The nephroid has a length of 24, and its area is 12 www.2dcurves.com /roulette/rouletten.html   (171 words)

 The Curves from Circles poster set The cardioid, limaçon and nephroid are roulettes obtained when one circle rolls along another. Light rays reflected from a curved mirror form a family of straight lines; the envelope of this family is a caustic curve. The cardioid and the nephroid are both caustics of a circle: if the light source is on the circle, a cardioid results; if the rays are parallel, a nephroid is formed. www.arbelos.co.uk /thecurvesfromcir.html   (218 words)

 Andrew Lawrence - Assignment 3   (Site not responding. Last check: 2007-10-15) Nephroid of Freeth:     r = 1 + 2 sin(t/2) The graph of r = e is, very similarly to r = 1, simply a circle with its center at the origin. And we established in our investigation of Freeth’s nephroid that r = sin(t) is a circle with radius 1 and center at [0.5, 90°]. web.pdx.edu /~lawaj/mth488/3.html   (824 words)

 nephroid of Freeth   (Site not responding. Last check: 2007-10-15) The curve, a botanic curve, is an extension of Freeth's nephroid (b = 2). This specimen is the strophoid constructed for a circle, named after the English mathematician T.J. Freeth (1819-1904). The Freeth's nephroid (b=2) can be used to construct a regular 7-sided figure, as follows: www.2dcurves.com /derived/strophoidn.html   (95 words)

 The Nephroid Family of Curves - National Curve Bank   (Site not responding. Last check: 2007-10-15) In addition to MATHEMATICA@, the Nephroid family of curves is easily entered and modified on a graphing calculator. The name nephroid means "kidney-shaped" from the Greek for nephróros meaning "kidney" and éidos meaning form. The nephroid was investigated extensively in the late 17th century. curvebank.calstatela.edu /nephroid/nephroid.htm   (271 words)

 Maybe this can forecast the market trend... best Nephroid   (Site not responding. Last check: 2007-10-15) Freeth's Nephroid -- from MathWorld Freeth's Nephroid -- from MathWorld A strophoid of a circle with the pole O at the center of the circle and the fixed point P on the circumference of the circle. The name nephroid (meaning 'kidney shaped') was used for the two-cusped epicycloid... Nephroid -- from MathWorld Nephroid -- from MathWorld The 2-cusped epicycloid is called a nephroid. www.ascot.pl /th/Fourier4/Nephroid.htm   (414 words)

 Nephroid   (Site not responding. Last check: 2007-10-15) The name nephroid means kidney shaped'' and was first used for the two-cusped Cardioid and reflected by it is a nephroid. Yates, R. Nephroid.'' A Handbook on Curves and Their Properties. www.itu.dk /edu/documentation/mathworks/math/math/n/n056.htm   (67 words)

 Optik am FB Physik der Philipps-Universität Marburg Being the caustic of a circle with light source at infinity, the nephroid approximates the cross-section of a hemispherical solar concentrator. The fact that the nephroid can be analytically described (x = a(3cos(t) - cos(3t)), y = a(3sin(t) - sin(3t)), from (2) again), is then an Application of Algebra to Optics. For our example of the nephroid, nonimaging optics mean that there isn't only one caustic caused by a point source at infinity sending out parallel light, but instead that there is a finite size sun, causing two caustics to be created in the hemispherical mirror. www.physik.uni-marburg.de /optik/nonimaging.html   (939 words)

 Little Gallery of Roulettes The 2:1 epicycloid is also known as the nephroid and was studied by Huygens, Tschirnhausen, and Bernoulli 1678-1692. First, here's a 7:1 epicycloid colored in lavender with three purple epiroulettes line drawn in the same picture, but with their drawing points beyond the radius of the rolling circle, so part of these curves lie inside the fixed circle. Next, here's a 2:1 epicycloid (that is, a nephroid) colored gold with a red epiroulette where the drawing point is within the radius of the rolling circle. aleph0.clarku.edu /~djoyce/roulettes/roulettes.html   (616 words)

 Name Your Poison   (Site not responding. Last check: 2007-10-15) Drinking a cup of coffee at an outdoor cafe one sunny day, DarkerOwl noticed there was a caustic nephroid in his java. Nephroid can be regarded as the caustic of a cardioid with radiant point at Nephroid can be also regarded as caustic of a circle for parallel rays www.magickalkingdom.com /smf/index.php?topic=8787.msg468291   (896 words)

 [No title]   (Site not responding. Last check: 2007-10-15) Special cases are: cardioid (b=a), nephroid (b=a/2)." epicycloidunitspeed::usage="s->epicycloidunitspeed[a,b][s] is a unit-speed parametrization of t->epicycloid[a,b][t]." epispiral::usage="t->epispiral[n,a][t] is an epispiral. The curve has n branches if n is odd and 2*n branches if n is even. The polar equation is r==a*E^(b*theta)." logspiralunitspeed::usage="s->logspiralunitspeed[a,b][s] is a unit-speed parametrization of t->logspiral[a,b][t]." nephroid::usage="t->nephroid[a][t] is a nephroid or kidney-shaped curve that is traced by a point on the circumference of a circle of radius a rolling around a fixed circle of radius 2*a. www.mathematik.hu-berlin.de /~gollek/seminarSS2002/alfredgray/CURVES.m   (2371 words)

 Special Epi- and Hypo- Cycloids Three of these are the Astroid, the Deltoid, and the Nephroid. Each of these can be generated in a different way, rather than as an Epi- or Hypocycloid. The Nephroid is an epicycloid with two points (i.e. www.lausd.k12.ca.us /NHHS_Highly_Gifted_Mag/mathweb/sketchpad/spcycloid.html   (259 words)

 Math Class. Programming.   (Site not responding. Last check: 2007-10-15) Each of the 32 points of division becomes the center of a circle whose radius is its distance from the highest point on the circle To construct Nephroid, the given circle is divided into 32 equal parts. Each of the 32 points of division becomes the center of a circle whose radius is its distance from vertical diameter. school.ort.spb.ru /library/vc98/futsch/math_pr2.htm   (89 words)

 Envelopes of Lines and Circles   (Site not responding. Last check: 2007-10-15) The nephroid is formed by an envelope of circles with centers on a given circle and each tangent to a diameter of the circle. Click here to obtain a GSP script for generating the nephroid. Next, the nephroid is outlined in two ways -- as an envelope of circles and as an envelope of lines. jwilson.coe.uga.edu /Texts.Folder/Envel/envelopes.html   (1005 words)

 [No title] To plot try Show[folium[-0.999,200],folium[-60,-1.001],PlotRange->{{-3,2},{-3,2}}]." freeth::usage="t->freeth[n,a][t] is Freeth's nephroid of order n. To plot try logspiralunitspeed[1,0.04][0,12Pi]." nephroid::usage="t->nephroid[a][t] is a nephroid or kidney-shaped curve that is traced by a point on the circumference of a circle of radius a rolling around a fixed circle of radius 2*a. A nephroid is a special case of an epicycloid. www.ma.umist.ac.uk /kd/mmaprogs/CURVES.m   (5260 words)

 nephroid - OneLook Dictionary Search Tip: Click on the first link on a line below to go directly to a page where "nephroid" is defined. Nephroid : A Glossary of Mathematical Terms [home, info] Phrases that include nephroid: freeth's nephroid, freeths nephroid, nephroid evolute, nephroid involute www.onelook.com /?w=nephroid   (146 words)

 Maybe this can forecast the market trend... best Nephroid Evolute   (Site not responding. Last check: 2007-10-15) the nephroid is the envelope of a set... mussel-shaped) (aka cochloid), nephroid (kidney-shaped), cycloid (circle, wheel... Nephroid Evolute -- from MathWorld Nephroid Evolute -- from MathWorld The evolute of the nephroid given by x = {{1\over 2}}[3\cos t-\cos(3t)] y = {{1\over 2}}[3\sin t-\sin(3t)] is given by x = \cos^3 t y = {{1\over 4}}[3\sin t+\sin(3t... www.ascot.pl /th/Fourier4/Nephroid-Evolute.htm   (382 words)

 Cynthia Schneider   (Site not responding. Last check: 2007-10-15) The curve Nephroid of Freeth with the polar equation r = 1 + 2 sin (t/2). If we replace the two in front of the sine function with a parameter b some interesting changes to our curve result. The following link is a graph of the Butterfly Curve using this equation. web.pdx.edu /~cynthias/Assignment_Three.htm   (180 words)

 toc The nephroid sliding on three symmetrically located lines A property of a chain of tangential circles Nephroid passing through the vertices of a regular triangle poncelet.math.nthu.edu.tw /chuan/zoo/toc.html   (155 words)

 Nephroid Generator   (Site not responding. Last check: 2007-10-15) Click with the mouse to send in a horizontal beam of light to reflect off the boundary of the circle. After you try a few beams at different heights, can you see the nephroid curve showing up as tangent to all the reflected lines? Superimpose the nephroid tangent to the reflected lines? www.geom.uiuc.edu /~fjw/calc-init/nephroid/ray.html   (60 words)

 Freeths   (Site not responding. Last check: 2007-10-15) In the picture above, O is the origin and P is the node where the curve crosses itself three times. If the line through P parallel to the y-axis cuts the nephroid at A then angle AOP is This can be used to construct a regular 7 sided figure. www-history.mcs.st-and.ac.uk /history/Curves/Freeths.html   (134 words)

 Results Page   (Site not responding. Last check: 2007-10-15) The name nephroid means "kidney shaped" and was first used for the two-cusped epicycloid by Proctor in 1878 (MacTutor Archive). In addition, Huygens showed in 1678 that the nephroid is the catacaustic of a circle when the light source is at infinity, an observation which he published in his Traité de la luminère in 1690... ; Yates, R. C. "Nephroid." A Handbook on Curves and Their Properties. eprints.bo.cnr.it /cgi-bin/show.pl?code=7705&arch=118   (173 words)

 Earliest Known Uses of Some of the Words of Mathematics (N) By it we mean an arbitrarily small but still always finite interval at both sides of the value x. NEPHROID was used for the bicuspid epicycloid by Richard Anthony Proctor (1837-1888) in 1878 in A treatise on the cycloid and all forms of cycloidal curves: According to E. Lockwood, in 1879 Freeth used the same name for another curve, Freeth's Nephroid. members.aol.com /jeff570/n.html   (5360 words)

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