Fractal program drawing types

Mandelbrot set

Description

Draws the Mandelbrot set.

Options

Iterations - Number of iterations to do before giving up (larger numbers mean more detail in the picture, but it takes longer to draw).

Notes

Since the Mandelbrot set is an index set for Julia sets, the index set controls work here.

Julia sets

Description

Draws Julia sets.

Options

Iterations - Number of iterations to do before giving up (larger numbers mean more detail in the picture, but it takes longer to draw).
Real coord - Julia sets arise from iterating z²+c. This option is the real part of c.
Imag coord - This option is the imaginary part of c.

Plot function

Description

Plots the graphs of functions.

Options

Step size - The program plots points just like you learned in grammar school, except it plots lots of them. A step size of -2 means that the distance between points plotted is 10^-2=.01. Decreasing the step size to -3, for example, will give more detail (distance between points would be .001), but it will take longer to draw.

Variables

The only variable is x.

Contour plot

Description

The color of each point is given by the function. It is a way to graph z=f(x,y), where the height, z, is indicated by the color.

Options

Gradient - Determines how much detail will be shown. For example, a small gradient may mean that function values between 0 and 1 are assigned light red, values between 1-2 are assigned a slightly darker red, etc. However, a large gradient would mean that perhaps values between 0 and .05 are assigned light red, values between .05 and .1 are assigned the slightly darker red color, etc. In other words, more colors are used over the same range of values.

Variables

The variables should always be x and y.

Formula fractal

Description

This iterates a complex function and stops when the iterates appear to converge to a point or infinity or, failing that, it stops when a certain fixed number of iterations have been done. It determines if they are converging by checking if the difference between the current iteration and the previous one is less than a specified tolerance. This is the index set part. It plays a similar role to the role that the Mandelbrot set plays to Julia sets.

Options

Iterations - Number of iterations to do before giving up (larger numbers mean more detail in the picture, but it takes longer to draw).
Initial real - The index set looks at what happens to a specific point. This is the real coordinate of that point.
Initial imag - The index set looks at what happens to a specific point. This is the imaginary coordinate of that point.
Tolerance - This was mentioned above. It determines how close the current and previous iterates have to be before we stop iterating. A tolerance of -5 corresponds to a tolerance of 10^-5=.00001. There is usually very little difference from changing this.
Initial type - The index set usually looks at what happens to a single point, but this option allows the point to vary. It is useful if you're not sure what point would be useful as a basis for your index set.
Gradient - This is similar to the gradient option for contour maps.

Variables

The variable is z. The real part of z is x, and the imaginary part is y. The index set parameter is c.

Notes

See the section (link) on index set controls. What's written here probably won't make a lot of sense without the mathematics (link) behind what we're doing.

Formula fractalJ

Description

Options

Iterations - Number of iterations to do before giving up (larger numbers mean more detail in the picture, but it takes longer to draw).
Initial real - The real part of the value c in the formula.
Initial imag - The imaginary part of the value c in the formula.
Tolerance - This was mentioned above. It determines how close the current and previous iterates have to be before we stop iterating. A tolerance of -5 corresponds to a tolerance of 10^-5=.00001. There is usually very little difference from changing this.
Gradient - This is similar to the gradient option for contour maps.

Variables

The variable is z. The real part of z is x, and the imaginary part is y. The index set parameter is c.

Notes

See the section (link) on index set controls. What's written here probably won't make a lot of sense without the mathematics (link) behind what we're doing.

Parametric plot

Description

Plots a 2d parametric curve. The curve is given by x=f(t), y=g(t) for some functions f and g.

Options

Step size - This tells how fast we increment t by, very similar to step size for function plotting.
Minimum t - The starting value of t.
Maximum t - The ending value of t.

Variables

The only variable is t.

Implicit plot

Description

Plots an implicit equation. All terms of the equation should be on one side. For example, to plot a circle of radius 3, the formula should be x^2-y^2-9.

Options

Step size x - This tells how fast we increment x by, very similar to the step size for function plotting.
Step size y - This tells how fast we increment y by, very similar to the step size for function plotting.

Variables

The variables are x and y.

Parametric plot 3d

Description

Plots a 3d parametric curve. The curve is given by x=f(t), y=g(t), z=h(t) for some functions f and g, h.

Options

Step size - This tells how fast we increment t by, very similar to the step size for function plotting.
Minimum t - The starting value of t.
Maximum t - The ending value of t.

Variables

The only variable is t. Use the 8, 9, 0 keys to rotate the image. Shift + 8, 9, or 0 rotates backwards.

Polar contour plot

Description

This is the same as contour plot, but the equation is given in polar. The color of each point is given by the function. It is a way to graph z=f(r,t), where the height, z, is indicated by the color.

Options

Gradient - Determines how much detail will be shown. For example, a small gradient may mean that function values between 0 and 1 are assigned light red, values between 1-2 are assigned a slightly darker red, etc. However, a large gradient would mean that perhaps values between 0 and .05 are assigned light red, values between .05 and .1 are assigned the slightly darker red color, etc. In other words, more colors are used over the same range of values.

Variables

The variables should always be r and t, (where t stands for theta).

Parametric surface

Description

Plots a 3d parametric surface. The curve is given by x=f(u,v), y=g(u,v), z=h(u,v) for some functions f and g, h.

Options

Step size u - This tells how fast we increment u by, very similar to the step size for function plotting.
Minimum u - The starting value of u.
Maximum u - The ending value of u.
Step size v - This tells how fast we increment v by, very similar to the step size for function plotting.
Minimum v - The starting value of v.
Maximum v - The ending value of v.

Variables

The variables are u and v. Use the 8, 9, 0 keys to rotate the image. Shift + 8, 9, or 0 rotates backwards.

Notes

The implementation of this is not great. There are some bugs that need to be worked out, and it should do hidden surface removal, but right now doesn't. You can use this to get a 3d plot of an ordinary function f(x,y) of 2 variables by setting x=u, y=v, and z=f(u,v).

Plot polar function

Description

Plots the graphs of polar functions.

Options

Step size - This tells how fast we increment t by, very similar to the step size for function plotting.
Minimum t - The starting value of t.
Maximum t - The ending value of t.

Variables

The only variable is t (stands for theta).

Polar implicit plot

Description

Plots an implicit equation given in polar coordinates. All terms of the equation should be on one side.

Options

Step size r - This tells how fast we increment r by, very similar to the step size for function plotting.
Step size t - This tells how fast we increment t by, very similar to the step size for function plotting.

Variables

The variables are r and t (stands for theta). Note that t is always between -π and π, so the results from this and plot polar function might not look the same.