mesh and surf for surfaces in MATLAB - Time & Space Complexity

Time Complexity: mesh and surf for surfaces

O(n^2)

Understanding Time Complexity

When we use mesh and surf in MATLAB, we create 3D surface plots from data grids.

We want to know how the time to draw these surfaces changes as the data size grows.

Scenario Under Consideration

Analyze the time complexity of the following code snippet.

% Create grid points
[x, y] = meshgrid(linspace(-5,5,n), linspace(-5,5,n));

% Calculate surface values
z = sin(sqrt(x.^2 + y.^2));

% Plot surface
surf(x, y, z);
    

This code creates a grid of points, computes a surface height for each point, and then plots the surface.

Identify Repeating Operations

Identify the loops, recursion, array traversals that repeat.

• Primary operation: Computing the surface values for each point in the grid.
• How many times: Once for each of the n x n points in the grid.

How Execution Grows With Input

As the grid size n increases, the number of points grows by n squared, so the work grows quickly.

Input Size (n)	Approx. Operations
10	100 (10x10)
100	10,000 (100x100)
1000	1,000,000 (1000x1000)

Pattern observation: Doubling n roughly quadruples the work because the grid is two-dimensional.

Final Time Complexity

Time Complexity: O(n^2)

This means the time to compute and plot the surface grows with the square of the grid size.

Common Mistake

[X] Wrong: "Increasing n only makes the code run a little slower because it's just one loop."

[OK] Correct: The code uses a grid with two dimensions, so the number of points grows by n times n, not just n.

Interview Connect

Understanding how plotting time grows with data size helps you write efficient visualizations and explain performance clearly.

Self-Check

"What if we used a 3D grid (x, y, z) instead of 2D? How would the time complexity change?"

Practice

1. What is the main difference between mesh and surf functions in MATLAB when plotting 3D surfaces?

easy

A. mesh creates a wireframe surface, while surf creates a colored surface.

B. mesh plots 2D graphs, while surf plots 3D graphs.

C. mesh only works with matrices, surf only works with vectors.

D. mesh colors the surface, surf shows only edges.

Solution

1. Step 1: Understand the function outputs

   mesh draws a grid-like wireframe of the surface without filling colors.

2. Step 2: Compare with surf

   surf draws a colored surface that fills the area between grid lines, showing height with color.

3. Final Answer:

   mesh creates a wireframe surface, while surf creates a colored surface. -> Option A

4. Quick Check:

   Wireframe vs colored surface = A [OK]

Hint: Remember: mesh = wireframe, surf = colored surface [OK]

Common Mistakes:

• Thinking mesh colors the surface
• Confusing 2D and 3D plotting
• Assuming mesh only works with vectors

2. Which of the following is the correct syntax to create a mesh plot of matrices X, Y, and Z in MATLAB?

easy

A. mesh(X, Y);

B. mesh(X, Y, Z);

C. mesh(X, Z, Y);

D. meshplot(X, Y, Z);

Solution

1. Step 1: Recall mesh function syntax

   The correct syntax for mesh requires three inputs: X, Y, and Z matrices representing coordinates.

2. Step 2: Check options

   Only mesh(X, Y, Z); matches the correct order and function name. mesh(X, Y); misses Z, B uses invalid function meshplot, C swaps Y and Z incorrectly.

3. Final Answer:

   mesh(X, Y, Z); -> Option B

4. Quick Check:

   Correct function and argument order = D [OK]

Hint: Syntax: mesh(X, Y, Z) with all three matrices [OK]

Common Mistakes:

• Using wrong function name like meshplot
• Swapping Y and Z arguments
• Omitting one of the required inputs

3. What will be the result of the following MATLAB code?

X = 1:3; Y = 1:3; [X, Y] = meshgrid(X, Y); Z = X.^2 + Y.^2; mesh(X, Y, Z);

medium

A. A colored surface plot with smooth shading.

B. A flat 2D grid with no height variation.

C. An error because meshgrid is used incorrectly.

D. A 3D wireframe surface showing a paraboloid shape.

Solution

1. Step 1: Understand meshgrid and Z calculation

   meshgrid creates matrices X and Y for grid points. Z is calculated as sum of squares, forming a paraboloid shape.

2. Step 2: Understand mesh plotting

   mesh plots a wireframe 3D surface of Z over X and Y, showing the paraboloid shape as a grid.

3. Final Answer:

   A 3D wireframe surface showing a paraboloid shape. -> Option D

4. Quick Check:

   meshgrid + paraboloid + mesh = wireframe paraboloid [OK]

Hint: meshgrid creates grids; mesh plots wireframe surface [OK]

Common Mistakes:

• Thinking mesh creates colored surface
• Assuming meshgrid returns vectors, not matrices
• Expecting 2D plot instead of 3D

4. Identify the error in this MATLAB code snippet:

X = 0:0.5:2; Y = 0:0.5:2; Z = X.^2 + Y.^2; surf(X, Y, Z);

medium

A. Z must be a matrix matching sizes of X and Y, not vectors.

B. surf requires only two inputs, not three.

C. X and Y must be row vectors, not column vectors.

D. The operator .^ is invalid for vectors.

Solution

1. Step 1: Check dimensions of X, Y, and Z

   X and Y are vectors, but Z is calculated element-wise and is also a vector, not a matrix matching grid points.

2. Step 2: Understand surf input requirements

   surf expects X, Y, Z matrices of the same size, usually created by meshgrid. Here, Z is not a matrix matching X and Y.

3. Final Answer:

   Z must be a matrix matching sizes of X and Y, not vectors. -> Option A

4. Quick Check:

   surf needs matching matrix sizes = C [OK]

Hint: Use meshgrid to create X, Y matrices before surf [OK]

Common Mistakes:

• Passing vectors instead of matrices to surf
• Misunderstanding element-wise operations
• Thinking surf accepts vector inputs directly

5. You want to plot the surface of the function z = sin(x) * cos(y) over the range x = -pi to pi and y = -pi to pi. Which MATLAB code correctly creates a colored surface plot using surf?

hard

A. x = -pi:pi; y = -pi:pi; Z = sin(x) * cos(y); surf(x, y, Z);

B. x = linspace(-pi, pi, 50); y = linspace(-pi, pi, 50); Z = sin(x) .* cos(y); surf(x, y, Z);

C. x = linspace(-pi, pi, 50); y = linspace(-pi, pi, 50); [X, Y] = meshgrid(x, y); Z = sin(X) .* cos(Y); surf(X, Y, Z);

D. X = -pi:pi; Y = -pi:pi; Z = sin(X) + cos(Y); mesh(X, Y, Z);

Solution

1. Step 1: Create grid points for x and y

   Use linspace to create vectors from -pi to pi with enough points, then use meshgrid to create matrices X and Y.

2. Step 2: Calculate Z values element-wise

   Calculate Z as sin(X) .* cos(Y) using element-wise multiplication to get matrix Z matching X and Y.

3. Step 3: Plot with surf

   Use surf(X, Y, Z) to plot the colored surface correctly.

4. Final Answer:

   x = linspace(-pi, pi, 50); y = linspace(-pi, pi, 50); [X, Y] = meshgrid(x, y); Z = sin(X) .* cos(Y); surf(X, Y, Z); -> Option C

5. Quick Check:

   meshgrid + element-wise .* + surf = B [OK]

Hint: Use meshgrid and element-wise .* for surface plots [OK]

Common Mistakes:

• Using * instead of .* for element-wise multiplication
• Not using meshgrid to create X and Y matrices
• Using vectors directly in surf without matching sizes