(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Prove that a line in 3D space is imaged to a line on the image plane in a pinhole camera model.

2. Relevant equations

A 3D point give by (X,Y,Z) will be imaged on the image plane at

x = f([itex]\frac{X}{Z}[/itex])

and

y = f([itex]\frac{Y}{Z}[/itex])

where f is the focal point.

3. The attempt at a solution

My first thought was a more intuitive one. If you have a line in 3D space and the point which is the pinhole, you have a plane on which both the pinhole and line lie. Where this plane intersects the image plane it forms a line and this is where the 3D line is mapped to on the image plane. However, this seems like too much hand waving.

Instead I decided to try saying that for the line in 3D there must be a parametric equation given by

X = at + d

Y = bt + e

Z = ct + g

Then

x = f[itex]\frac{at+i}{ct+k}[/itex]

y = f[itex]\frac{bt+j}{ct+k}[/itex]

From here I know that for a line to exist on the image plane there must be a q and m such that

y = mx + q

Yet, this doesn't seem to lead me in the right direction.

Any suggestions? Thank you for your time!

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# Computer Vision Geometry - Collinear Points In A Pinhole Camera Model

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