## Understanding the function surf() in Octave/MATLAB

I’ve been studying Machine Learning for a few days now, on Coursera. After finishing 1^{st} and 2^{nd} weeks, I’ve got the first assignment for the course where you are supposed to implement Linear Regression using Gradient Descent.

While trying to do the assignment, I’ve come across with a function, `surf()`

, for which I’ve struggled ~~a few hours~~ a little bit to understand.

What was confusing for me in the assignment was this line below:

Because of the way meshgrids work in the

`surf()`

command, we need to transpose`J_vals`

before calling surf, or else the axes will be flipped.`J_vals = J_vals';`

I couldn’t get that at first glance, so I’ve started fiddling with the function and plots until a few hours later I’ve found this explanation from the docs^{1}.

If

`x`

and`y`

are vectors, then a typical vertex is`(x(j), y(i), z(i,j))`

. Thus, columns of z correspond to different x values and rows of z correspond to different y values.

*Take home message: Always check for the docs FIRST.* ¯_(ツ)_/¯

Anyway, to visualize this I’ve plotted a graph using the code below:

```
A = [1 2 3 4 5 6];
B = [7 8 9 10 11 12];
C = magic(6);
% Matrix C:
% 35 1 6 26 19 24
% 3 32 7 21 23 25
% 31 9 2 22 27 20
% 8 28 33 17 10 15
% 30 5 34 12 14 16
% 4 36 29 13 18 11
surf(A, B, C)
```

By the way, I should mention that in the assignment we were assigning the elements of the matrix C (which corresponds to `J_vals`

in the snippet below) like:

```
% theta0_vals is a 1x100 row vector, which in our case is the matrix A
% theta1_vals is a 1x100 row vector, which in our case is the matrix B
for i = 1:length(theta0_vals)
for j = 1:length(theta1_vals)
t = [theta0_vals(i); theta1_vals(j)];
J_vals(i,j) = computeCost(X, y, t);
end
end
```

In the above code, we are assigning the result of the computation from A_{i} and B_{j} to C_{i,j}.

So you can think of it as C_{i,j} = (A_{i}, B_{j}).

Considering the explanation above, after using `surf(A, B, C)`

you might think that `C(A(1), B(3))`

which is `C(1,9)`

should be equal to C_{1,3} which is 6.

But as you can see from the above surface plot, the point `(1, 9)`

maps to 31 or you can use `interp2(A, B, C, 1, 9)`

which will give you the same result.

Hence, we take the transpose of the matrix C:

`surf(A, B, C')`

to get :