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I would realize the papercraft of a geodesic sphere like THIS .
Is the dual of the one discussed in THIS OTHER QUESTION .
Where can I find the printable nets, or the online resources to create them?
In the other discussion I learned that there are 3 classes of possible tessellations.
One of these classes needs less pentagons to tessellate the sphere?
I don't exactly know how much faces I need, indicatively between 100 and 200.
Which chord factors I should consider?
Thanks to all
genna,
The link that I provided in the former discussion...
http://thomson.phy.syr.edu/thomsonapplet.php
...provides for display of a dual. Unfortunately, the java applet will only export a data set for the geodesic triangular tessellation; not the dual.
The dual can be constructed from the triangular tessellation, by importing the vertex (point) data set (from the java applet) into a 3D-graphics program, and then following the rules of the definition of a dual. (Each pent/hex face will use the triangular vertex point as the centerpoint of a circle, being equidistant from the sphere center and normal to the central vector. The intersections of these circles will produce the linear edges of the dual faces.)
All tessellated duals, based on the icosahedron, will have the same number of pentagonal faces (12), regardless of how many hex faces are produced.
-Taff
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I have a 3D mesh I'm calculating vertex normals for, from the mesh's face normals. Each vertex normal is computed as an average of all the normals of the faces that share that vertex.
All that works fine, except in cases where -- due to uneven subdivision of faces over the surface of the mesh -- vertex normals can be skewed. Here is an example of an object with a skewed vertex normal, in one of its corners:
In this image you can see the various face normals (blue), the ideal vertex normals (yellow), and the problematic vertex normal (red) which is being skewed by the many face normals on the heavily subdivided side of the mesh which all share the vertex.
So my question is: is it possible to calculate a vector median, instead of an average? Here's an extra image to further illustrate:
The idea is to weight the face normals by the angle of the edges that connect to the vertex in question, before adding them to that vertex's normal.
So in my first image, the large grouping of normals on the right would have the same cumulative weight as the single normal on the top/left....resulting in an overall ideal normal for that corner vertex.
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Given a graph of a formula, how can I find its formula?
Do I need some OCR or something like that?
In general you need:
detect the graph position (bounding box) in image
you are usually looking for rectangle outlined (usually in black) area
extract/input graph scales
either input them by user or read via OCR (as you suggested). The scale values and units are usually along the x,y axises of the graph bounding box. You need to find at least 2 separate points per axis to form transformation from pixels to units. In case of non linear scales you need to have more points to detect log,etc scales ...
extract graph points
the graph curve is usually in different color or stroke width so it should be relatively easy to extract all points. Form a list of points (in case of function sorted by x).
apply curve fitting on the points from #3
there are many approaches ... can use piecewise interpolation, piecewise approximation BEZIER/SPLINE, interpolation polynomials (Lagrange...) or FFT/DCT (sin waves). For non complicated shapes approximation search minimizing distance between points and computed polynomial will be enough. This step mainly depends on what precision, output function form, runtime you need.
Hard to be more specific as you did not provide any details what so ever.
Finding the mathematical algorithm to which matches an input and output together might help a bit.
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I have searched a lot of google, reviewed my old mathematics books, But I am stuck why can't we find area of an irregular shape from its perimeter? If we have find perimeter from area, Is that possible to reverse the process and find area from perimeter.
This process is working fine for square and rectangle but not for totally irregular shapes.
I'm asking this question because we have one field with irregular shape, and we want to know its area, We can measure its 5 sides.
Don't we have any method to find area from perimeter for irregular shapes.
Any help will be appreciated.
Consider a rope and form an irregular shape. Measure it's area. Now change the shape you just formed with the rope. Measure the area again. Is it the same? Not necessarily.
Or consider a square and a rhomb (which is not a square) with the same perimeter. Do they have the same area? No. Therefore, how do you think could you find out what its area is if you don't have the information of the shape?
You cannot get the area if you just have the information of the perimeter, but don't know anything else.
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I have a problem it is like trig and pre calc. We are given vector points for OA which is (6 0) and vector OB (-6 0) and OC (5 root11) and they are all points on a circle with a radius of 6. How exactly do you determine or show that these points lie on the circle?
For any three points which are not mutually co-linear, there is a point equidistant from each member of the set. Finding this point given three arbitrary points (and not given the radius) is a much more informative project if you're learning trigonometry.
In your example, the solution is simple: x2 + y2 = r2. Each of the points you provided have values such that the sum of their squares equals 36, as expected for a circle of radius 6. Your problem is especially easy because the circle in question is already centered on the origin.
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Imagine that we have a regular plane defined in three-dimensional Euclidean space ("world space").
On this plane, we trace an arc from point A to point B, like a bullet trajectory, in the plane's space ("local space"). That is to say, if the plane was transformed in "world space" using for example rotation, the arc would still remain relative to the plane as it is defined in the plane's space.
We now apply an arbitrary deformation to the plane. It follows that the arc would likewise be deformed in world space.
Which branch of geometry (or mathematics in general) would be used to approach calculating the path of this arc in world space; and for problems like these in general; and how would this problem be approached in software?
Update:
This falls within the domain of Differential geometry.
I believe you are looking for affine transformations, though if your arbitrary deformations to the plane are "strong enough" you might be working with Projective geometry instead.