Calculating rotation of equally spaced items tangent to spiral - math

Question

I'd like to programmatically draw a shape like this where there is an underlying spiral and equally spaced objects along it, placed tangent to the spiral as shown in this sketch:
I found an example of how to determine equally spaced points along the spiral here and am now trying to place hemispheres along the spiral. However, I'm not sure how to calculate the angle the shape needs to be rotated.
This is what I have so far (viewable here):
var totalSegments = 235,hw = 320,hh = 240,segments;
var len = 15;
points = [];
function setup(){
createCanvas(640,480);
smooth();
colorMode(HSB,255,100,100);
stroke(0);
noFill();
//println("move cursor vertically");
}
function draw(){
background(0);
translate(hw,hh);
segments = floor(totalSegments);
points = getTheodorus(segments,len);
angles = getAngles(segments, len);
for(var i = 0 ; i < segments ; i++){
let c = color('blue');
fill(c);
noStroke();
// draw shape
if(i % 2){
// console.log(i, ' ', angles[i]);
// try rotating around the object's center
push();
// translate(points[i].x, points[i].y)
rotate(PI/angles[i]);
arc(points[i].x, points[i].y, len*3, len*3, 0, 0 + PI);
pop();
}
// draw spiral
strokeWeight(20);
stroke(0,0,100,(20+i/segments));
if(i > 0) line(points[i].x,points[i].y,points[i-1].x,points[i-1].y);
}
}
function getAngles(segment, len){
let angles = [];
let radius = 0;
let angle = 0;
for(var i =0; i < segments; i++){
radius = sqrt(i+1);
angle += asin(1/radius);
angles[i] = angle;
}
return angles;
}
function getTheodorus(segments,len){
var result = [];
var radius = 0;
var angle = 0;
for(var i = 0 ; i < segments ; i++){
radius = sqrt(i+1);
angle += asin(1/radius);
result[i] = new p5.Vector(cos(angle) * radius*len,sin(angle) * radius*len);
}
return result;
}

Answer 1

Note that your drawing shows Archimedean spiral while link refers to Theodorus one.
Archimedean spiral is described by equation in polar coordinates (rho-theta)
r = a + b * Theta
where a is initial angle, b is scale value (describes distance between arms), r is radius.
And angle Theta + Pi/2 describes normal to spiral in point at parameter Theta
If you need an approximation to divide spiral into (almost) equal segments - use Clackson formula (example here)
theta = 2 * Pi * Sqrt(2 * s / b)
for arc length s