Math problems, homogeneous coordinates?
Hi - Basically, homogeneous coordinates are another way to express coordinates of varying numbers of dimensions. When you're working with Cartesian (2-dimensional) coordinates, here's the deal: *Homogeneous coordinates will have three numbers and take the form (x, y, w). *The two Cartesian values X and Y correspond to the homogeneous values x/w and y/w respectively. This also means that, in converting Cartesian coordinates to homogeneous ones, w = 1. So to answer your first question, (2,3),(-1,0), and (7,0) would become (2, 3, 1), (-1, 0, 1), and (7, 0, 1) respectively. Note that, depending on your instructor, course, and applications, you may be instructed to use colons [(2:3:1), etc.] instead of commas. And for your third question, (7,3,-4),(1,1,1), and (0,-2,2) become (-7/4, 3/4), (1,1), and (0,-1) respectively. I'm sorry there isn't enough time left for me to answer the rest of your questions, but if you repost them I'll be happy to try and help. In the meantime, see the two links below for more about homogeneous coordinates and their applications. Hope this helps!
Why is a homogeneous coordinate system needed in a transformation matrix?
Usually you see homogeneous coordinates system used where projection is expected. Like in computer vision, working with cameras that project 3D world points into 2D pixel coordinates.According to this document the reason for working with homogeneous coordinates:• Simpler formulas• Fewer special cases• Unification and extension of concepts• Duality
What does homogeneous in "homogeneous coordinate" refer to?
Homogeneous coordinates for a point in space aren’t unique. The coordinates [math](a,b,c)[/math] and the coordinates [math](\lambda a, \lambda b, \lambda c)[/math] represent the same point.So, if you have a function that’s defined over points in the space and you’re using homogenous coordinates, then your function needs to be a homogeneous function, otherwise it’s not uniquely determined by the point. That’s where the name comes from—they’re homogeneous coordinates because they’re coordinates used in homogeneous functions.Homogeneous functions are those where multiplying all of the inputs by a scalar results in the output being multiplied by a power of the same scalar—[math]f(\lambda a, \lambda b, \lambda c) = \lambda^\alpha f(a,b,c)[/math].I’m not certain, but I think homogeneous functions are named as a generalization of homogeneous polynomials, which are polynomials in which all terms with a non-zero coefficient have the same degree, which would explain the term “homogeneous”.
Calculate the x coordinate of the centroid for the homogeneous body of revolution shown.?
Ok I have the answer to the problem. Including the solution. My question is what formula or steps are being used to get x1 = 2 - (3(2)/8) = 1.25... Basically I'm lost in getting x1 x2 and x3. I know how to plug in the rest of the numbers and how to get v1 v2 and v3, but its the x part that I'm lost in. I saw that for a hemisphere you can use 3r/8 and if plug that in i get (3(2)/8) which is x1 but why does x1 have a (2 -) in front. Anyone that can explain with details I would really appreciate it. Thanks.
Find the coordinates of the center of gravity.?
1) A contributes nothing: I = Σ mr² I = 200g * ( (10cm)² + (10cm√2)² + (10cm)² ) = 80 000 g·cm² = 0.008 kg·m² 2) B and D contribute nothing; A and C are d = 10cm / √2 away. I = 100g * (10cm/√2)² + 200g * (10cm/√2)² = 15 000 g·cm² = 0.0015 kg·m²
Statics: x coordinate of centroid?
I don't know the "method of parts" but the centroid is 2/3 the distance from the pointy side or 1/3 the distance from the tall side if that makes any sense. It depends on which way the triangle is pointed. |\ |.\ |...\ |.....\ --------- 36" in this case its 36(1/3) = 12" from the left ....../| ...../.| ..../..| ...----- 36 in this case its 36(2/3) = 24" from the left.