Analytic geometry for imaging
Points and vectors, dot and cross products, lines and planes, ray intersection and linear interpolation: the calculations behind lighting, rendering and gradients.
- 29 cards
- 5 topics
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A few cards from the deck
Dot product: what is it?
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The number obtained by adding the products of the matching components of two vectors; it also equals ‖u‖ ‖v‖ cos θ, where θ is the angle between them.
It is positive for an acute angle, zero for a right angle, negative for an obtuse one: it is the basic orientation test in rendering.
What is u · v for u = (2, 3) and v = (3, −2), and what does the result tell you?
0: the vectors are perpendicular0: the vectors are parallel12: the vectors form an acute angle6: the vectors form an acute angle
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0: the vectors are perpendicular2 × 3 + 3 × (−2) = 6 − 6 = 0. A zero dot product means perpendicular vectors, not parallel ones: for parallel vectors it equals ± ‖u‖ ‖v‖.
12comes from a sign mistake on the second component,6from forgetting the second one.The dot product of u and v equals ‖u‖ ‖v‖ × …, and it is zero when the two vectors are ….
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The dot product of u and v equals ‖u‖ ‖v‖ × cos θ, and it is zero when the two vectors are perpendicular.
cos 90° = 0: this is how a right angle is tested without computing the angle. To get θ itself, take the arccosine of u · v / (‖u‖ ‖v‖).
Cross product: what is it?
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An operation on two 3D vectors that gives a vector perpendicular to both, of length ‖u‖ ‖v‖ sin θ, oriented by the right-hand rule.
Unlike the dot product, the result is a vector, and the order matters: u × v = −(v × u).