Our Demo for the day.

We would like to move the camera in space.

Remember the camera has three properties

A current location, specified in modeling coordinates
A Look At point, specified in modeling coordinates
An up vector.

In the new code.

NewView( camera, at, up) {
    var eye = lookAt(camera, at, up);
    var proj =  this.Frustum(-1, 1, -1, 1, 2, 20);
    proj = mult(proj,eye);
    this.gl.uniformMatrix4fv(this.projLoc, false,flatten(proj));
}

Note, the frustum is fixed.
Messing with the frustum will change the view dramatically.

So To move in space I need to change the values of the camera.

I am going to go to a different coordinate system called spherical coordinates

This is a way to represent 3 space.
A distance and two angles are used.
The first angle is the angle from the x axis in the xy plane.
The second angle is the angle in from that point in the yz plane.
The distance is the radius of the sphere on which we want the point.

In my mind, this is an extension of polar coordinates.

$x = r cos(\theta) sin(\phi)$
$y = r cos(\theta) sin(\phi)$
$z = r cos(\phi)$

We will swap the role of y and z.

I am reasonably confident that I have a sign error, but this will work.

My general plan is to

Keep the camera position in Cartesian coordinates in modeling space
Keep the angles $ \theta, \phi$ as variables.
To rotate
- Increment or decrement $ \theta, \phi$ as needed.
- Recalculate the at position.
When I want to move.
- Calculate an offset from the current position.
- Update the camera position by that offset
- Recalculate the at position.

Note in this world,

I step to where I am looking.
I step the same amount ahead of me that I am looking.
I don't need to do that, but that is probably your problem, not mine.

So how do I compute the new "target" position

Pretend the camera is at the origin.
Using $ \theta, \phi$ and a fixed step size, pretend they are spherical coordinates.
Add this offset to the current camera position to find the new camera position.
Add this offset to the new camera position to find the new at position.

I really should update the up vector

If up and at are the same, I have a problem.