Overview of Graph Types

At the top of every plot window there are three popup buttons that set the type of the currently selected graph. Firstly the preimage and image spaces (which maps to a non-empty cell in the table below), then the coordinate system and thirdly the visualization/projection mode, which is described below.

To
ℝ²ℝ³
F
r
o
m
Re, Im, Abs, Phase, Image, Riemann-Image---
-GraphParametricParametric
ℝ² -GraphImage, VectorfieldParametric
ℝ³ -Implicit-Vectorfield
--ParametricParametric
S¹ × S¹ ---Parametric

Spherical coordinates

On ℝ² we have cartesian (x and y) and polar coordinates (r and angle φ, which is counterclockwise from the positive real axis). On ℝ³ we have cartesian (x,y,z), spherical (r, φ, θ) and cylindrical coordinates (r, φ, z).

CPlot follows these conventions: In spherical coordinates φ is the counterclockwise angle between the point projected into the XY-plane and the positive real axis, θ is the angle between the Z-axis and the line from 0 to the point in question (when θ = π/2, r and φ are the same as polar in ℝ²).

For cylindrical coordinates: r is the distance from the point to the Z-axis, φ the angle between the point projected into the XY-plane and the positive real axis, and z the z-coordinate of the point. For z=0, this is the same as polar again.

Real-Valued Functions

CPlot draws the two most common real graph types: Line graphs for functions ℝ → ℝ and surface graphs for ℝ² → ℝ. To draw these, select ℝ → ℝ or ℝ² → ℝ from the first dropdown box.

Variable names on ℝ²: x, y, r = √(x²+y²).

Variable names on ℝ: x, y = 0, r = |x|, t = x if t is not used as a parameter or function name..

For consistency with the other modes there is also z = x+iy, u = x, v = y.

ℝ → ℝ

ℝ² → ℝ

On ℝ³ CPlot can draw the zero set {x ∈ ℝ³ | f(x) = 0}, also known as isosurface, level set or implicit function graph.

Variable names are: x, y, z, r = √(x²+y²+z²) and again u = x, v = y.

In the points display mode, this draws points where f(x,y,z) ≤ 0.

x⁴+y⁴+z⁴ - (x²+y²+z²) = -0.42

Complex Functions

The graph of a complex function can either be projected from the 4-dimensional space ℂ² down into ℂ × ℝ, which is what the Real Part, Imaginary Part, Absolute Value and Phase modes do, or we only plot the image, either on ℂ or on the Riemann sphere.

Variable names: z, x = re(z), y = im(z), r = abs(z) and the synonyms u = x, v = y.

Imaginary part of Γ(z)

Riemann image of a pole

Image mode for exp(z), using texture

Parametric Functions

These are the images of functions from {ℝ, S¹, ℝ² or S¹ × S¹} to {ℝ² or ℝ³}. All preceding plot modes are special cases of these. In the S¹ modes, the variables x and y will have values in [0, 2π], where f(0) and f(2π) will be glued together. In the other modes they can be set to any interval (on the right of the window, under "Axis").

Variable names: x and y a.k.a. u and v. r and z as for graphs. y will be 0 on ℝ and S¹. For functions on ℝ or S¹ there is also t as synonym for x as long as t is not used as a parameter or function name.

Logarithmic spiral

"Spherical Product" from the gallery

Vector Fields

Functions from ℝ² → ℝ² or ℝ³ → ℝ³ can be plotted as vector fields, which have four scaling modes:

  1. Unscaled: Draws the vectors in the axis scale.
  2. Normalized: Scales it so that the largest visible vector has about the same length as the grid width.
  3. Direction Only: Draws all vectors with the same length, again about grid width long.
  4. Connected: Draws connected field lines. It scales the vectors like Normalized does.

Variable names: x, y, z. Also u, v, r as usual (r being √(x²+y²+z²) on ℝ³) and z = x+iy on ℝ² as usual.

 

2D vector field

3D vector field

Phase of Gamma(z) as vector field

Plotting More Than One Function

There are three axis types: 2D, 3D and the Riemann sphere. Combining 2D and 3D yields a 3D plot where the 2D graphs are embedded in the XY-plane, except for ℝ → ℝ graphs, which get embedded in XZ. For different embeddings, parametric modes like ℝ → ℝ³ can be used.

Combining the sphere mode with anything else gives an error ("Axis type mismatch").

Graphs can be added and removed with the +/- buttons next to the "Graphs" box in the right hand side of the window. The list entries work like radio buttons in that exactly one of them is always selected (written in bold), which is the current graph whose definition and settings can be modified.

sin and cos

Real parts of complex sin and cos