Engineering
Proof that the vector from the viewpoint of a pinhole camera to the vanishing point (in the image plane) of a set of 3D parallel lines is parallel to the direction of the parallel lines:Let L be a set of parallel 3D lines, and let v be their vanishing point in the image plane. Let O be the viewpoint of the camera. We want to prove that the vector from O to v is parallel to the direction of the lines in L.Consider two lines l1 and l2 in L. Let P1 and P2 be two points on these lines. Let IP1 and IP2 be the interpretation planes of these lines passing through O. Since the lines are parallel, the interpretation planes are also parallel. Let l be the line of intersection of the interpretation planes, passing through O. Let Q1 and Q2 be the projections of P1 and P2 onto the image plane, respectively. Let v be the vanishing point of the lines in L. Then, Q1Q2 is parallel to the lines in L and passes through v. Let R1 and R2 be the intersections of IP1 and IP2 with the image plane, respectively. Then, R1R2 is parallel to Q1Q2 and passes through O. By the similar triangles formed by the image plane, the interpretation plane, and the object plane, we have:|OR1|/|OQ1| = |OR2|/|OQ2|.Since R1R2 is parallel to the lines in L, and Q1Q2 is parallel to the image plane, we have:|OR1|/|OQ1| = |R1R2|/|Q1Q2|.Therefore, |R1R2|/|Q1Q2| = |OR2|/|OQ2|.Since Q1Q2 is parallel to L, and R1R2 is the intersection of the image plane and the interpretation planes of l1 and l2, we have: |R1R2|/|Q1Q2| = |P1P2|/|L|,where |L| is the length of the segment between P1 and P2 on the lines in L.Therefore, |P1P2|/|L| = |OR2|/|OQ2|.Since this equation holds for any two points P1 and P2 on the lines in L, we conclude that the vector from O to v is parallel to the direction of the lines in L.
\o you sell them? "Fivepence farthing for oneTwopence for two, the Sheep replied. "Then two are cheaper than one? Alice said in a surprised tone, taking out her purse. "Only you MUST eat them both, if you buy two, said the Sheep. "Then I'll have ONE, please, said Alice, as she put the money down on the counter. For she thought to herself, "They mightn't be at all nice, you know. The Sheep took the money, and put it away in a box: then she said "I never put things into people's handsthat would never doyou must get it for yourself. And so saying, she went off to the other end of the shop, and set the egg upright on a shelf. "I wonder WHY it wouldn't do? thought Alice, as she groped her way among the tables and chairs, for the shop was very dark towards the end. "The egg seems to get further away the more I walk towards it. Let me see, is this a chair? Why, it's got branches, I declare! How very odd to find trees growing here! And actually here's a little brook! Well, this is the very queerest shop I ever saw! The tone of this passage is best described as unusual and strange. logical and orderly. realistic and ordinary. negative and critical.