get an angle between 2 points and rotate a point about an other point with this angle - C++

Question

I'm basically trying to create 2D lines based on points from bezier curves. All the points of the bezier curves are well placed and everything seems in order. Starting with these points I'm creating 2 other points on the z axis which will be the border of the line :

    glm::vec3    p1 = pos[i];
    p1.z = p1.z + (size / 2);

    glm::vec3    p2 = pos[i];
    p2.z = p2.z - (size / 2);

Then I change these points positions by rotating them around the main point :

pm is the mobile point rotating around the fix point pf

    glm::vec3   rotP = glm::vec3(0.0f, 0.5f, 0.0f);

    float   co = cos(angle);
    float   si = sin(angle);

    // CLOCKWISE
    rotP.x = (pf.x - pm.x) * co + (pf.z - pm.z) * si + pm.x;
    rotP.z = -(pf.x - pm.x) * si + (pf.z - pm.z) * co + pm.z;

angle is the angle between the backward and forward point on the bezier curve :

depForward is x, glm::vec3(1.0f, 0.0f, 0.0f)

    glm::vec3 normForwardUnit = normalize(p2 - p1);
    float angle = (acos(dot(depForward, normForwardUnit)));

The problem that I get is that the rotations are wrong. Some of my lines are correct but it seems to depend on the orientation of the lines.

not correct example correct example

I think the problem comes from the format of the rotation but I'm still unable to understand. I tried to normalize the angle to different ranges :

//0 to 2PI
    if (angle < 0) { angle += 2 * PI; }
//-PI to PI
    if (angle > PI)        { angle -= 2 * PI; }
    else if (angle <= -PI) { angle += 2 * PI; }

Other ways to calculate the angle :

    float angle = atan2(p2.z - p1.z, p2.x - p1.x);

To rotate the points counter-clockwise :

//COUNTER CLOCKWISE
    rotP.x = (pf.x - pm.x) * co - (pf.z - pm.z) * si + pm.x;
    rotP.z = (pf.x - pm.x) * si + (pf.z - pm.z) * co + pm.z;

I'm not sure why you're trying to do rotations here. It seems to me like all you want is to create points that give your line thickness. That would be achieved by generating at a set distance _perpendicular_ to your line. You can generate a perpendicular vector using the cross product. By crossing `p2 - p1` with the Z-axis, you'll get a perpendicular vector. Or just switch the X/Y values and negate one of them. Same thing. Then normalize it, multiply by +/- the thickness. These are the offsets of your border points from the line. — paddy, Dec 14 '22 at 20:53
Thanks @paddy, indeed it's a good way to do what I wanted. Here's the code to complete your approach : _backP and nextP being the point before and the point after_ ' // VEC FORWARD VECTOR glm::vec3 forwardVec = normalize(backP - nextP); //PERPENDICULAR VEC glm::vec3 perpVec = cross(forwardVec, glm::vec3(0.0f, 1.0f, 0.0f)); perpVec = normalize(perpVec); //MID POINT glm::vec3 midP = midPoint(backP, nextP); glm::vec3 p1 = midP + (width * perpVec); glm::vec3 p2 = midP - (width * perpVec);' — UnchartedLine, Dec 15 '22 at 15:41

score 1 · Accepted Answer · answered Dec 16 '22 at 08:07

In case anyone needs it, here's the implementation of paddy's approach. You could use the point between backP and nextP instead of midPoint to place your new points.

backP and nextP being the point before and the point after of the b curve

  // VEC FORWARD VECTOR
  glm::vec3 forwardVec = normalize(backP - nextP);

  //PERPENDICULAR VEC
  glm::vec3 perpVec = cross(forwardVec, glm::vec3(0.0f, 1.0f, 0.0f));
  perpVec = normalize(perpVec);

  //MID POINT
  glm::vec3 midP = midPoint(backP, nextP);

  // GEN POINTS
  glm::vec3 p1 = midP + (width * perpVec);
  glm::vec3 p2 = midP - (width * perpVec);

Spektre · Answer 2 · 2022-12-16T07:53:18.483

I think you should definately look at this:

Is it possible to express "t" variable from Cubic Bezier Curve equation?

In case you insist on your way you do not need to use any angles nor rotations...

You have line p0,p1 which is sampled from your polynomial curve so:

tangent = p1-p0

However its better to have better approximation of tangent so either take it by 1st derivation of your curve or use 2 consequent lines (p0,p1) , (p1,p2) then tangent at point p1 is:

tangent = p2-p1

For more info see:

How do i verify the gradient at midpoint coordinate which i calculate by using cubic bezire curve equation

Now take bitangent (z axis of your camera which can be extracted from camera matrix) and use cross product to get normal

normal = normalize(cross(tangent,binormal))

now you just displace the p1 by normal:

p1'  = p1 + 0.5*curve_thickness*normal
p1'' = p1 - 0.5*curve_thickness*normal

do the same for all points of your curve ... after that you just render quads using p' and p'' points ...

However with this approach you might run into problems that might need further tweaking see:

draw outline for some connected lines

get an angle between 2 points and rotate a point about an other point with this angle - C++

2 Answers2