algebraic calculation 1 Prove that the vector from the viewpoint of a pinhole camera to the vanishing point (which is a point on the image plane) of a set of 3D parallel lines in space is parallel to the direction of that set parallel lines. Please show steps of your proof.

**Hint:** You can either use geometric reasoning or algebraic calculation.

If you choose to use geometric reasoning, you can use the fact that the projection of a 3D line in space is the intersection of its “interpretation plane” with the image plane. Here the interpretation plane (IP) is a plane passing through the 3D line and the center of projection (viewpoint) of the camera. Also, the interpretation planes of two parallel lines intersect in a line passing through the viewpoint, and the intersection line is parallel to the parallel lines.

If you select to use algebraic calculation, you may use the parametric representation of a 3D line: P = P0 +tV, where P= (X,Y,Z)^{T} is any point on the line (here ^{T} denote for transpose), P0 = (X0,Y0,Z0)^{T} is a given fixed point on the line, vector V = (a,b,c)^{T} represents the direction of the line, and t is the scalar parameter that controls the distance (with sign) between P and P0.

algebraic calculation 2. Show that relation between any image point (xim, yim)^{T} (in the form of (x1,x2,x3)^{T} in projective space ) of a planar surface in 3D space and its corresponding point (Xw, Yw, Zw)^{T} on the plane in 3D space can be represented by a 3×3 matrix. You should start from the general form of the camera model (x1,x2,x3)^{T} = M_{int}M_{ext}(Xw, Yw, Zw, 1)^{T}, where the image center (ox, oy), the focal length f, the scaling factors( sx and sy), the rotation matrix R and the translation vector T are all unknown. Note that in the course slides and the lecture notes, I used a simplified model of the perspective project by assuming ox and oy are known and sx = sy =1, and only discussed the special cases of a plane. So you cannot directly copy those equations I used. Instead you should use the general form of the projective matrix, and the general form of a plane n_{x} X_{w} + n_{y} Y_{w} + n_{z} Z_{w} = d.

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