• ### What The Hare Said To Hector

Hector: Alright, explain your game to me, Mr. Hare. Hare: First, I will stand somewhere along the race track, but I won’t tell you where, and this wall hides me from view. Now, do you see this large contraption here? Hector: Indeed. There is a -meter long net suspended from a high cable that extends […]

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Jan 20th, 2012 | Filed under Uncategorized
• ### Miss Register, I Presume?

In my AIM 2010 paper, I describe how to obtain a projective transformation from the image plane to the laser plane in a line laser 3D range imaging system. With the laser oriented vertically (i.e. perpendicular to the transport direction of the object being scanned), this allows for mapping of image coordinates directly to 3D […]

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Jun 4th, 2011 | Filed under Uncategorized
• ### Rise of the Quaternions

Adolphus finally quit messing around and started using a quaternion representation for rotations internally! The quaternion class itself is simple, and conversion to and from rotation matrix and axis-angle representations is fairly straightforward. The magic happens in converting from Euler angles — all twelve valid conventions! By solving the conversion to quaternion for all twelve […]

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Oct 18th, 2010 | Filed under Uncategorized
• ### Rotating Lines

Problem: Given a line of slope m in the Euclidean plane, what is the slope m’ of the line rotated (counterclockwise) by angle θ? Solution: Suppose we have an equation for the line of the form y = mx + b. We can ignore b as it is unrelated to the slope (in effect, we […]

Jun 28th, 2010 | Filed under Uncategorized
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• ### So Close, Yet So Far Away

I humbly entreat any analytical intellects of greater constitution than my own (of which, to be sure, there is no dearth) to enlighten me in matters mathematical. First of all, if I have a 5-dimensional space which consists of a 3-dimensional Euclidean space plus direction (defined by inclination and azimuth) — that is, the set […]

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Jun 15th, 2010 | Filed under Uncategorized
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• ### Snap To Grid

I like Gabor Herman‘s definition of discrete Euclidean space. He defines, for any positive real number and any positive integer : This chops N-space up into a square (cubic, etc.) Bravais lattice, with primitive vectors of magnitude . Each point in the discrete space is then associated with a primitive cell (or Voronoi neighbourhood, in […]

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Mar 17th, 2010 | Filed under Uncategorized
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You have a right-handed Cartesian coordinate basis of a three-dimensional Euclidean space, with axes , , and . You’re given some spatial-directional vectors of the form , where and are, respectively, the inclination angle (from the positive -axis zenith) and the azimuth angle (measured right-handed from the positive -axis) of an associated direction, which is […]

Mar 14th, 2010 | Filed under Uncategorized
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• ### On Path and Edge Strengths in Fuzzy Graphs

How does one define the strength of a path in a fuzzy graph? Mathew and Sunitha state that “the degree of membership of a weakest [edge] is defined as [a path’s] strength” without any apparent justification. Saha and Udupa mention that several measures, including sum, product, and minimum, all seem plausible, but also ultimately choose […]

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Oct 17th, 2009 | Filed under Uncategorized
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• ### Distributed Descriptive Statistics and Magic

Here’s an interesting problem. Anyone know of a solution to this? There are 24 wizards in the land of Network. They live spread out far from one another, because if too many of them got too close, the concentration of magical power would create a singularity and destroy the universe. They are all aware of […]

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Oct 10th, 2008 | Filed under Uncategorized
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Posts Tagged ‘math’