National
          Science Foundation
National Curve Bank

A Collection of Famous Plane Curves

created in POVRAY (Persistence Of Vision RAY tracing)

Deposit #73

Patrick J. Gleason




lemniscate
The Joy of Mathematics
Color-coded Curves

In each of the pictures below, x is the horizontal dimension (width), y is the vertical dimension (height), and z is the color (depth). To more clearly show the contours of the z surface, the color has been quantized (summarized) into 16 distinct levels:

0 = BLACK 1 = BROWN 2 = RED-BROWN 3 = RED
4 = ORANGE 5 = YELLOW-ORANGE 6 = YELLOW 7 = YELLOW-GREEN
8 = GREEN 9 = CYAN 10 = BLUE 11 = INDIGO
12 = VIOLET 13 = BLUE-GREY 14 = GREY 15 = WHITE

The drawing program plots thin black lines to represent the x and y axes, and a thin white line to show where the 3-dimensional surface crosses the z=0 plane. You can click on any picture on this page to see a full-screen version that you can study in detail.

Circle

Circle
r = cos(q)
z = x/r - r
Drawing Limits: (-2, -2) - (2, 2)
Min z: -3.3, Max z: 0.993744

Cardioid

Cardioid
r = 1 + cos(q)
z = 1 + x/r - r
Drawing Limits: (-2, -2) - (2, 2)
Min z: -2.3, Max z: 1.99374

Pascal's Rose

Pascal's Rose
r2 = cos2(2q)
z = (x4 - 2x2y2 + y4)/r4 - r2
Drawing Limits: (-2, -2) - (2, 2)
Min z: -6.176, Max z: 0.99996

Three Leaf Limaçon

Three Leaf Limaçon
r = cos(3q)
z = (4x3/r3 - 3x/r) - r
Drawing Limits: (-2, -2) - (2, 2)
Min z: -3, Max z: 0.993743

Daisy

Daisy
r2 = cos2(4q)
z = ((x8 - 12x6y2 + 38x4y4 - 12x2y6 + y8) / r8) - r2
Drawing Limits: (-2, -2) - (2, 2)
Min z: -5.53901, Max z: 0.999956

Checkerboard

Checkerboard
z = cos(x) + cos(y)
Drawing Limits: (-10, -10) - (10, 10)
Min z: -1.99981, Max z: 2

Two Dimensional Sampling Function

Two Dimensional Sampling Function
z = (sin(x)/x)(sin(y)/y)
Drawing Limits: (-10, -10) - (10, 10)
Min z: -0.217229, Max z: 1

Rays

Rays
z = cos(3q)
z = 4x3/r3 - 3x/r
Drawing Limits: (-10, -10) - (10, 10)
Min z: -1, Max z: 1

Four Petal Daisy

Four Petal Daisy
r2 = cos2(4q), with odd lobes suppressed
z = x/r - r
Drawing Limits: (-2, -2) - (2, 2)
Min z: -3.3, Max z: 0.993744

Six Leaf Clover

Six Leaf Clover
r2 = cos2(3q)
z = 16x6/r6 - 24x4/r4 + 9x2/r2 - r2
Drawing Limits: (-2, -2) - (2, 2)
Min z: -6.1261, Max z: 0.993755

Two Petal Rose

Two Petal Rose
r = cos(2q)
z = ((x2 - y2) / r2) - r
Drawing Limits: (-2, -2) - (2, 2)
Min z: -3.3, Max z: 0.993744

Spiral

Spiral
r = q
z = tan-1(y/x) - r
Drawing Limits: (-2, -2) - (2, 2)
Min z: -5.13848, Max z: 3.13479

Butterfly

Butterfly
basically, r = versin(5q) with lobe sizes adjusted
z = ((4y5 - 4y3x2 + 8yx4) / r5) - (3y3 - 3yx2)/r3 - r
Drawing Limits: (-2, -2) - (2, 2)
Min z: -5.11643, Max z: 2.86601

Circus Tent

Circus Tent
a pair of polynomials with 5 roots each
z = x5 - 5x3 + 4x - y5 + 5y3 - 4y
Drawing Limits: (-2, -2) - (2, 2)
Min z: -6.91267, Max z: 6.88453

Two Hyperbolae and a Circle

Two Hyperbolae and a Circle
z = (x2 - y2 - 1) (y2 - x2 - 1) (x2 + y2 - 1)
Drawing Limits: (-1.5, -1.5) - (1.5, 1.5)
Min z: -5.07812, Max z: 1.67009

Radial Hyperbolae and Circle

Radial Hyperbolae and Circle
r2 = (x2 - y2 - 1) (y2 - x2 - 1) (x2 + y2 - 1)
z = (x2 - y2 - 1) (y2 - x2 - 1) (x2 + y2 - 1) - r2
Drawing Limits: (-1.1, -1.1) - (1.1, 1.1)
Min z: -1.42419, Max z: -0.851859

Intersecting Waves

Intersecting Waves
basically, y = sin(3x) and x = sin(5y)
z = (sin(3x) - y) (sin(5y) - x)
Drawing Limits: (-3, -3) - (3, 3)
Min z: -9.50629, Max z: 8.48362

Black Hole

Black Hole
has only imaginary roots
z = x2 - xy + y2
Drawing Limits: (-2, -2) - (2, 2)
Min z: 0, Max z: 9.21882

A Lonely Pulse

A Lonely Pulse
basically, y = sin(x)/x
z = 10 - (10sin(x) / x) - y
Drawing Limits: (-20, -20) - (20, 20)
Min z: -14.931, Max z: 27.1723

Vertical Bars

Vertical Bars
z = 10cos(x)
Drawing Limits: (-20, -20) - (20, 20)
Min z: -9.99919, Max z: 10

Zig Zag

Zig Zag
basically, y = cos(x) + x
z = 5cos(x) + 2x - y
Drawing Limits: (-20, -20) - (20, 20)
Min z: -52.8971, Max z: 58.1044

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References and more Curves