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 | |||||
---|---|---|---|---|---|
ℂ | ℝ | ℝ² | ℝ³ | ||
F r o m | ℂ | Re, Im, Abs, Phase, Image, Riemann-Image, Color, Riemann-Color, Histogram | - | - | - |
ℝ | - | Graph | Parametric | Parametric | |
ℝ² | - | Graph, Implicit* | Image, Vector Field, Color | Parametric | |
ℝ³ | - | Implicit* | - | Vector Field | |
S¹ | - | - | Parametric | Parametric | |
S¹ × S¹ | - | - | - | Parametric |
*Contour plots can be done by plotting sin(πf) in implicit mode (which draws contours on all integers).
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.
Spherical coordinates
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), phi = arg(x+iy).
Variable names on ℝ: x, y = 0, r = |x|, t = x.
For consistency with the other modes there is also u = x, v = y, z = x on ℝ (for switching between real and complex types), z = 0 on ℝ² (for switching between ℝ, ℝ² and ℝ³). All of these automatic variables can be overridden by parameters and zero-ary functions with the same name (for all graph types).
ℝ → ℝ
ℝ² → ℝ
On ℝ² and ℝ³ CPlot can also draw the zero set {x ∈ ℝⁿ | f(x) = 0}, a.k.a. isosurface, level set or implicit function graph.
Variable names on ℝ³ are: x, y, z, r = √(x²+y²+z²), phi, theta (spherical coordinates) 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
x⁴+y⁴ - (x²+y²) = -0.28
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, on ℂ, on the Riemann sphere or by color coding.
Variable names: z, x = re(z), y = im(z), r = abs(z), phi = arg(z) and the synonyms u = x, v = y.
Imaginary part of Γ(z)
The image mode, especially when using textures, assigns a color value to every point in the preimage rectangle, maps that rectangle and draw the points at f(z) with the color from z. This shows directly how some area is moved, stretched and folded by the function. The image can be drawn onto the plane representation of ℂ or onto the Riemann sphere (tends to work better when the function has poles).
Image mode for exp(z), using texture
Riemann image of the pole of 1/z²
The color modes assign a color value to every point in the complex plane (now as the image range!) and draw the point at z (now in the preimage range) with the color at f(z). There are four ways of assigning the color:
The difference between the image and color modes is that image mode draws points at f(z) with the color at z, while color mode draws points at z with the color at f(z).
Color mode with tiled texture
The same function with phase texture in Riemann mode
Histograms take some number (controlled by the quality setting) of random points, map them to f(z) and approximate the resulting probability distribution by splitting ℂ into bins (size controlled by the grid setting) and counting how many values get mapped into each bin. The volume of each bar then shows the approximate probability density at that point. In points mode it just plots the mapped points, so the density of the points in some area is related to the volume of the matching histogram bar. Three source distributions are available:
Riemann-Histogram for z+2
These are the images of functions from {ℝ, S¹, ℝ² or S¹ × S¹} to {ℝ² or ℝ³}. Most other 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 left of the window, under "Axis").
Variable names: x and y a.k.a. u and v. r, phi 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
Functions from ℝ² → ℝ² or ℝ³ → ℝ³ can be plotted as vector fields, which have four scaling modes:
Variable names: x, y, z. Also u, v, r ( = √(x²+y²+z²) on ℝ³), phi, theta (on ℝ³) and z = 0 on ℝ² as usual.
2D vector field
3D vector field
Phase of Gamma(z) as vector field
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 of the "Graphs" box in the left 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.
Graphs can be hidden and unhidden with the checkboxes in the graphs list.
sin and cos
Real parts of complex sin and cos