vector - swift: orient y-axis toward another point in 3-d space

Question

Suppose you have two points in 3-D space. Call the first o for origin and the other t for target. The rotation axes of each are alligned with the world/parent coordinate system (and each other). Place a third point r coincident with the origin, same position and rotation.

How, in Swift, can you rotate r such that its y-axis points at t? If pointing the z-axis is easier, I'll take that instead. The resulting orientation of the other two axes is immaterial for my needs.

I've been through many discussions related to this but none satisfy. I have learned, from reading and experience, that Euler angles is probably not the way to go. We didn't cover this in calculus and that was 50 years ago anyway.

Answer 1

Got it! Incredibly simple when you add a container node. The following seems to work for any positions in any quadrants.

// pointAt_c is a container node located at, and child of, the originNode
// pointAtNode is its child, position coincident with pointAt_c (and originNode)

// get deltas (positions of target relative to origin)
let dx = targetNode.position.x - originNode.position.x
let dy = targetNode.position.y - originNode.position.y
let dz = targetNode.position.z - originNode.position.z

// rotate container node about y-axis (pointAtNode rotated with it)
let y_angle = atan2(dx, dz)
pointAt_c.rotation = SCNVector4(0.0, 1.0, 0.0, y_angle)

// now rotate the pointAtNode about its z-axis
let dz_dx = sqrt((dz * dz) + (dx * dx))

// (due to rotation the adjacent side of this angle is now a hypotenuse)
let x_angle = atan2(dz_dx, dy)
pointAtNode.rotation =  SCNVector4(1.0, 0.0, 0.0, x_angle)

I needed this to replace lookAt constraints which cannot, easily anyway, be archived with a node tree. I'm pointing the y-axis because that's how SCN cylinders and capsules are directed.

If anyone knows how to obviate the container node please do tell. Everytime I try to apply sequential rotations to a single node, the last overwrites the previous one. I haven't the knowledge to formulate a rotation expression to do it in one shot.