Calculating Vanishing Points in Pinhole Camera Photography: Exploring Perspective Projection and Image Formation

Q: Given a pinhole camera whose focal length is 2mm. Calculate the coordinates of the vanishing point of all the parallel lines in the real world which are parallel to the line passing through the point (0.1m, 0.3m, 0.1m) and the pinhole.

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A:

To calculate the coordinates of the vanishing point of all lines parallel to the line passing through the point (0.1m, 0.3m, 0.1m) and the pinhole of a pinhole camera, follow these steps:

1. Determine the Direction Vector: The direction vector 𝑣 of the line can be derived from the given point to the pinhole (assumed at the origin (0,0,0)). This gives 𝑣=(0.1,0.3,0.1).
   
   
2. Projection onto the Image Plane: The focal length 𝑓 of the pinhole camera is given as 2mm, which should be converted to meters for consistency in units, i.e., 𝑓=0.002 meters.
   
   
3. Scaling the Direction Vector to Find Intersection with the Image Plane: The image plane is at 𝑧=𝑓.
   We need to find the scaling factor 𝑘 such that the 𝑧-component of the scaled direction vector 𝑘𝑣 equals the focal length 𝑓. This is calculated by setting the 𝑧-component of 𝑘𝑣 equal to 𝑓:
   
   𝑘𝑣 = 𝑓
   
   
   𝑘×0.1 = 0.002
   
   𝑘 = 0.002/0.1 = 0.02
   
   
   
4. Calculate the Intersection Coordinates: Apply the scaling factor to each component of 𝑣:
   
   𝑘𝑣=0.02×(0.1,0.3,0.1)=(0.002,0.006,0.002)

The 𝑥 and 𝑦 components at the point where the line intersects the image plane are the coordinates of the vanishing point. Thus, the coordinates on the image plane for the vanishing point are (0.002,0.006) meters, or converting to millimeters for practical purposes, (2,6) mm.