Famous Curves and Their Tangents

Jen-chung Chuan

NTHU, Taiwan

e-mail: jcchuan@math.nthu.edu.tw

webpage: http://sylvester.math.nthu.edu.tw/d2/talk-cabriworld-2004/


The MacTutor History of Mathematics archive (http://www-history.mcs.st-and.ac.uk/history/) contains an interesting section devoted to “famous curves”, many of which were studied before the publication of René Descartes' La Géométrie. The development of mathematics has since linked Algebra and Geometry so strongly that plane curves are introduced nowadays not in terms of their geometric shapes but the equations instead. In this study we shall demonstrate how some of the interesting famous curves, along with their tangents, can be constructed synthetically. According to our method, the appropriate algebraic formulation of the curves is chosen. The steps of the construction are then designed based on the analytic process. Finally Cabri Geometry is used to visualize the interpretation of algebraic formulae. The construction has the advantage that

1) it is exact;

2) no infinite process is involved;

3) nothing is hidden. 


Kappa curve

r = cot t

Kampyle of Eudoxus

r = sec 2 t

inverse of the quadrifolium

r = sec 2t

double folium

r = sin t sin 2t

Right Strophoid

r = sec t cos 2t


r = - sec t cos 2t


r = cos 3t


r = cos 2t

Freeth's nephroid

r = 1 + 2 sin(t/2)

conchoid of Nicomedes

r = b + a sec t

Cayley's sextic

r = cos 3 t/3


r = sec t - 4cos t

trisectrix of Maclaurin

r = sec t/3

trisectrix of Catalan

r = sec3 t/3

limacon of Pascal

r = a cos t + b

cissoid of Diocles

r = sin t tan t

conchoid of de Sluze

r = k2 cos t / a + a sec t

ellipse x = a cos t
y = b sin t
hyperbola x = a sec t
y = b tan t
bullet nose x = a cos t
y = b cot t
cross curve x = a sec t
y = b csc t
Lissajous x = cos 3t
y = sin 4t
Chebyshev polynomial x = cos t
y = cos 3t
lemniscate of Gernono x = cos t
y = sin t cos t
witch of Agnesi x = tan t
y = cos 2 t
piriform x = 1+sin t
y = a cos t (1+sin t)
serpentine curve

x 2y + a2y - bx = 0

butterfly curve

x6 + y6  = x2

falling sand

y2 (1 - x2) =1

folium of Descartes

x 3 + y 3 - 3xy = 0


y2 =  x4 - x6


y2 =  x2 (1- x2)2

Visual Index


1. Famous Curves Index: http://www-history.mcs.st-and.ac.uk/history/Curves/Curves.html
2. Cabri Geometry, http://www.cabri.net/index-e.html
3. J. Dennis Lawrence, A Catalog of Special Plane Curves
4. The Purple Bear's Web, http://purplebearsweb.co.uk/Fly/Knots/Why_collections_of_small_thing/why_collections_of_small_thing.html