CS112:: 3D visualization and working with recursion

 

Plotting 3D surfaces

Sometimes it is useful to plot 3D surfaces. Here is an example of a simple mathematical equation.

        z = x2 - y2

How to plot this equation:

  1. Create two vectors, say -10<=x<=10 and -8<=y<=8.
  2. Create two new vectors that contain the squares of the x and y vectors from Step 1.
  3. Use meshgrid to create matrices of all the x squared and y squared values in proper orientation.

    The effect of meshgrid is to create a matrix X with the x-values along each row, and a matrix Y with the y-values along each column. This is done so that it is easy to evaluate a function z=f(x,y) of two variables on the rectangular grid. Click here for an example of how meshgrid works.

  4. Create the matrix Z in terms of X and Y (where Z(i,j) = X(i)^2 - Y(j)^2)

 

Changing viewpoint

MATLAB tracks view using azimuth and elevation. azimuth is a polar angle in the x-y plane, with positive angles indicating counterclockwise rotation of the viewpoint (see diagram below). You can think of it as the horizontal rotation about the z-axis, as measured in degrees from the negative y-axis. Elevation is the angle above (positive angle) or below (negative angle) the x-y plane. You can see it as the vertical elevation of the viewpoint in degrees

Here is an example of viewing an object from directly overhead:

az = 0;
el = 90;
view(az, el);

Click here for an assortment of general 2D and 3D plotting examples.

Recursion

In this problem we will review recursion and write a couple of recursive MATLAB functions.

  1. Write a function called power()to compute x^n, according to the following recursive definition:

    x^0 = 1.0

    for all x

    x^n = x * x^(n-1)

    for n>0
    2. Test your function.
  2. Write one more function, called morePower(), to compute x^n according to the following definition:

    x^n = x^(n/2) * x^(n/2)

    for even n values

    x^n = x * x^(n/2) * x^(n/2)

    for odd n values
    3. Test your morePower() function.
  3. Write a function that takes a string as input, and returns true if the input string consists only of capital (english) letters, false otherwise. Test your function.
  4. Define a function, capitalize(), that takes a string as input, and returns it with all letters in capital case.
Here is the link to your cs112 assignment:

Assignment 8