编程代考 Q1: In the 3D space, assuming that the line passes through the point (x1, y

Q1: In the 3D space, assuming that the line passes through the point (x1, y1, z1) and (x2, y2, z2), please describe the formula of the line in 3D space.

Assuming: -;-; -;
Vector Form

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(,,)=(,,)+ (a,b,c)
Parametric Form

Symmetric Form

Q2: Show that the perspective projection of a line in 3D is still a line on the image plane.

As indicated in Solution to Q1, each point = (, , ) of a line in 3D can be represented as

where and are the slope, and and are the intercept. After perspective projection with focal length equal to f, assuming is the corresponding point of point in the image plane, then based on the similar triangle rule, we have

, y = (2)

Based on equation (1) and (2), we have

– (4)

As such, we have

As we can see it represents a line with slope and intercept .

Q3 Given a 3D scene point P=(0, 2, 2) and a 3×4 projection matrix (intrinsic parameter and extrinsic parameters) as following

Calculate the 2D image coordinate of the scene point after camera projection

When we multiply the projection matrix with the Homogeneous coordinates of the scene point, we have the corresponding Homogeneous coordinates of the 2D image coordinate
As such, the final 2D image coordinate is (-3.5, 10).

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