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Using atan2 with offset circle

I am trying to calculate the Arc Length, so that I can color my circle border given the Arc Length.
Users can click on the circle edge and the code should automatically calculate the Arc Length from position y=radius x=cx. Given the function atan2(), I cannot achieve this successfully, because if I understand correctly, atan2 return the angle between positive X-axis and the ray from my center of circle towards the point clicked. But I need the angle between the point clicked and the y-axis.
I am attaching the picture, in case it will make more sense:
What I have currently is:
// Calcualte the deltas for point X and Y which are required for atan2 function
let deltaX = event.pageX - this.centerX
let deltaY = event.pageY - this.centerY
// The Math.atan2() function returns the angle in the plane (in radians) between the positive x-axis and the ray from (0,0) to the point (x,y), for Math.atan2(y,x)
let angleRadian = Math.atan2(deltaY, deltaX)
// Apply formula to calculate Arc length
let arcLength = radius * angleRadian
I am aware that atan2 calculates the angle as I described above, but I do not know how to modify my code to achieve what I need it to do - that is, get the angle between the Point Clicked and the y-axis.

Just swap atan2 arguments to get angle from OY axis.
For counterclockwise (CCW) system:
let angleRadian = Math.atan2(deltaX, deltaY)
(based on formulas cos(a)=sin(Pi/2-a), sin(a)=cos(Pi/2-a))
For clockwise (CW) system, where OY axis direction is down:
let angleRadian = Math.atan2(deltaX, -deltaY)

JavaScript - rotate a shape by a fixed angle

I want to move a shape around the circumference of a circle on HTML canvas. I am using the following JavaScript logic:
speed = 0.005;
angle = Math.PI/2;
angle += speed * direction;
var x = cx + (radius * Math.cos(angle));
var y = cy + (radius * Math.sin(angle));
direction is set by a key press (left arrow = -1, right arrow = +1). cx and cy are fixed - they are the x and y co-ordinates of the center of the circle around which the shape is moving.
I want to move the shape around the circle in fixed steps, like the seconds hand of a clock. However, there should be 187 steps. I know that dividing 360/187 = 1.9251 degrees = 0.03359 radians. However, my drawing function is inside a loop, so writing angle += 0.03359 makes the shape spin around the circle forever.
How can I make each key press move the shape either clockwise or anti-clockwise around the circle, but in steps of 0.3359 radians?
I am using the logic I found in the answer written by rafaelcastrocouto for this question: how to move object in circle forward and backward in html5 canvas?

You need to save the starting angle in avariable that you access each time you begin to draw:
This line:
angle = Math.PI/2;
should then look like this:
angle = window.starting_angle;
On each keypress, you either increment or decrement this variable and redraw the shape.

Specify origin of Math.atan2

Math.atan2() is a very useful function for calculating angles. However, I cannot wrap my head around one thing:
$(document).mousemove(function(event){
r = Math.atan2(event.pageY, event.pageX);
deg = r * 180/Math.PI;
console.log(deg);
})
console.log indicates that 0,0 from where the angle is being calculated is at the upper left corner of the screen. How would I go about calculating the angle from a different origin, say the centre of the screen?

You would subtract the coordinates of your origin from the coordinates that you want to find the angle of:
r = Math.atan2(event.pageY - originY, event.pageX - originX);

What's the significance of 1/cos(x) in this code for a 3d canvas game?

I've been experimenting with HTML5 canvases lately and came across this 3d example with relatively little code behind it. I was hoping to find a good introduction to 3d rendering, but I'm having more trouble understanding the geometry behind the code than I was expecting to. I set up a JSbin and copied over the code that was used on his website to play with. I'm stuck at understanding the meaning of
deltaX=1/Math.cos(theta);
which is later used in:
if (deltaX>0) {
stepX = 1;
distX = (mapX + 1 - x) * deltaX;
}
else {
stepX = -1;
distX = (x - mapX) * (deltaX*=-1);
}
Source
My best guess is that it's used for the relation cos(x) = adjacent/hypotenuse in a right triangle, but I don't understand where the triangle would fit in, if at all.

If you draw a line from the origin (0, 0) with direction theta (measured from the x-axis), then
deltaX = 1/cos(theta) is the distance on this line until the vertical line x = 1 is met, and
deltaY = 1/sin(theta) is the distance on this line until the horizontal line y = 1 is met.
It is indeed a triangle relation. In the first case, the triangle has the points (0, 0), (1, 0) and the point (1, y) where the line meets the vertical line x=1.
(mapX, mapY) is a grid point with integer coordinates, and (x, y) is a point in the square [mapX, mapX+1) x [mapY, mapY+1).
distX computes the distance of the next vertical grid line in theta-direction, and distY the distance of the next horizontal grid line.
Remark: The computation fails if the direction is a multiple of π/2, i.e. the direction is exactly right, up, left, or down, because sin(theta) = 0 or cos(theta) = 0 in that case. This probably does not happen in your program, because the playerDirection starts with 0.4 and is incremented or decremented by 0.07.

Detecting mouse courser when is close to polyline

I have a programming question with some math weight. I have a map with shapes(polylines) drown on it. I can take the screen coordinates of that shapes and translate them to map coordinates and reverse. I am capturing mouse position and moving around the map. How can I recognize if I come in proximity to another shape drown on the map while I am moving the mouse. I was thinking to create a radius of points around the mouse cursor, then constantly looping trough available shapes (I imagine I can load their coordinates in arrays) for a match. However that will be very slow I think. The point is that when I am in proximity (for example 15px) I will snap the muse position to that close shape. Any suggestions?

Now - if you really want to make it perfect - you can calculate the distance of a cursor to each line segment.
For each line segment (defined by points D and E)
Calculate line formula for segment DE in format:
Ax + By + C = 0
A = D.y - E.y
B = E.x - D.x
C = (plug in point D) = -1 * (A * D.x + B * D.y)
Now plug in your cursor position to the formula:
A * cursor.x + B * cursor.y + C = YOUR DISTANCE TO THE LINE SEGMENT
*One thing - this is distance to the unbounded line. You now want to make sure that you are between the two segment points. So make sure the angles in your cursor, D, E triangle are all < 90 degrees. A number of ways to do that, look into the dot product formula to learn a fast one.
Now if anlges are less than 90, use the distance to the line, else, use the min distance to either point of segment (D & E). Now you have a complete snap to lines functionality.

If you have every point / line segment of the shapes (which you should with the polylines), here is a possible quick and simple routine:
For each shape
Figure center of shape by averaging each constituent point (i.e. - for a pentagon, add all five point.x up, divide by 5 - do same for all point.y). Call this average shape.x and shape.y. Use distance formula to figure proximity to your mouse. (Mouse.x - Shape.x)^2 + (Mouse.y - Shape.y)^2... you don't have to take the square root of that since you are only interested in the closest shape.
Keep track of the minimum distance "squared" for each shape. The minimum is your closest shape center.
If you want snap to behavior with a maximum range, just also make sure the distance squared is < pixel distance squared.
If you want to make it very effiecient, the shape centers do not need to be constantly refigured, just calculated once. The center will scale and translate the same as any other point if you are converting between screen space and other coordinates as you mentioned.