Find camera pose by knowing triangle coordinates

Question

I have this concrete problem where I know the 3 coordinates of a triangle $ABC$ in a $xyz$ frame reference and I need to estimate the position $P$ and direction $vec{n_{p}}$ of a stationary camera with known focal distance $f$. For this purpose I have the image from the camera where I know $H$, $W$ and the respective 2D positions $a$, $b$, $c$ in pixels.

My questions:

Is this possible by knowing only 3 points $ABC$?
Does make difference if triangle $ABC$ is equilateral or not since I know the map $ABC => abc$?
How to estimate $P$ and $vec{n_{P}}$ relative to frame $xyz$?

I read already Camera Pose Estimation from Vanishing Points, Step by Step Camera Pose Estimation for Visual Tracking and Planar Markers and How to compute camera pose from Homography matrix? but sincerily I couldn't realize how to solve my problem.

Cedron Dawg · Answer

No.  No.  You can't.
Not completely at least.  With only three points, the math has two solutions.  Your triangle could be tilted in one of two positions.  Like picking a square root.
From a single frame, you might be able to tell and measure motion from a blur gradient, but that is difficult.
From multiple frames, calculating the position of the camera for each frame, the velocity is simply a matter of differentiation techniques on a finite sequence.

Find camera pose by knowing triangle coordinates

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