I'm doing a 2D water simulation using springs, and interpolating between each spring-end when rendering to create a smooth water surface. The resulting water looks a little bit more like jelly. Is it possible to modify the catmull-rom to make 'pointy curves', or do I need to use a different algorithm. I just can't think of the correct search terms for what I want.
Example of what I'd like to recreate
(source: vimeocdn.com)
Instead of something like this
Cheers
You want the surface of the water to form cusps: points where the tangents on either side are not identical. The two endpoints of the Catmull-Rom spline control the tangent, so you can probably get some pointy waves by offsetting these higher.
Related
i was wondering if anyone of u here knows how to smooth a polygon in Maya? I've tried 2 methods which i found online. One of which is 'Vertice Averaging' and the other 'Smooth' which are both under the 'Mesh' option.
Vertice Averaging caused my polygons to have 'gaps' or 'holes' between the triangles, which i do not intend for that to happen.
While Smooth causes the polygon's face to have 4 vertex instead of the original 3, which i do not want as well, as i need a polygon with triangle faces.
http://img.photobucket.com/albums/v483/dragonlancer/PolygonAveragingampSmoothing.jpg
And to whoever told me that it is a bug, i tried, but doesnt work =[
You said you wanted to maintain your tris so you could switch the smooth option 'Add Divisions' from exponential to linear.
If you're getting gaps, its because the original mesh has verts which are not welded together. Try Edit Mesh > Merge with a small tolerance value before running average or smooth.
In general you'll get more pleasant results if you smooth a quadrangular mesh instead of a trimesh - when you subdivide quads the results are very similar to NURBS curves, whereas smoothed triangles always tend to look look like old-fashioned 1990's game graphics.
I have a general question about what method to use for smoothing a 3D (xyz) grid.
My program has large matrixes of 3D points obtained with a stereovision method. The shape of the result is always something like a semisphere, but it has a rugosity due to stereovision errors I want to eliminate.
The question is, how to do it? Rigth now I have half developed a method for soomthing, but I think there may be a better method.
My actual idea is to use Hermite method. The idea is to:
Take all XY and smooth in two directions ->XYnew and XnewY
Convert the Hermite lines into Bezier lines and find the cross point between XYnew and XnewY, having the new point. (Repeat with all points, normally 2000)
Use hermite XYZ smoothing having XYZnew.
Rigth now I have the hermite surface smoothing and hermite line smoothing inplemented in C++, but the middle part is not as easy as espected.
Anyway, my question is, is this a correct method or is there another one which may be better?
Of course the idea is to elliminate the error generated by the stereovision method, this is not a computer graphics problem, is more a data treatment problem
Appendix:
At first I thougth that with a Z smoothing would be suficient, but clearly it is not, there is also a lot of XY error. In the images below you can see the Z fitting working but still it is really rugous as it can be seen in the 2 image. (The colours are deformations and shoul be quite continous)
Unless you have better priors, it's hard to beat the classic Taubin's algorithm: http://mesh.brown.edu/taubin/pdfs/taubin-iccv95a.pdf
Mapping a point cloud onto a 3D "fabric" then flattening.
So I have a scientific dataset consisting of a point cloud in 3D, this point cloud comprises points on a surface that is curved. In order to perform quantitative analysis I however need to map these point clouds onto a surface I can then flatten. I thought about using mapping tools sort of like in the case of the 3d world being flattened onto a map, but not sure how to even begin as I have no experience in cartography and maybe I'm trying to solve an easy problem with the wrong tools.
Just to briefly describe the dataset: imagine entirely transparent curtains on the window with small dots on them, if I could use that dot pattern to fit the material the dots are on I could then "straighten" it and do meaningful analysis on the spread of the dots. I'm guessing the procedure would be to first manually fit the "sheet" onto the point cloud data by using contours or something along those lines then flattening the sheet thus putting the points into a 2d array. Ultimately I'll probably also reduce that into a 1D but I assume I need the intermediate 2D step as the length of the 2nd dimension is variable (i.e. one end of the sheet is shorter than the other but still corresponds to the same position in terms of contours) I'm using Matlab and Amira though I'm always happy to learn new tools!
Any advice or hints how to approach are much appreciated!
You can use a space filling curve to reduce the 3d complexity to a 1d complexity. I use a hilbert curve to index lat-lng pairs on a 2d map. You can do the same with a 3d space but it's easier to start with a simple curve for example a z morton order curve. Space filling curves are often used in mapping applications. A space filling curve also adds some proximity information and a new sort order to the 3d points.
You can try to build a surface that approximates your dataset, then unfold the surface with the points you want. Solid3dtech.com has the tool to unfold the surfaces with the curves or points.
Is it possible to calculate intermediate points of a curve...Here is my mirror image
In the above image.Is it possible to calculate the intermediate points(one side) by knowing starting and ending point
If you know something about the curve it is, and it all depends on what you know about the curve (start and end points, initial slopes, center points, etc). There are generally two approaches:
If you know the equation of the curve, it's possible to do this exactly. Commonly curves like this are either circles or Bezier curves, and if you know it's either of these, you can fit all the other points exactly just given a few.
You can also do a cubic spline fit. This is a standard approach to fitting smooth curves so packages to do this are very common. On a smooth curve like this, give then end points, and, say, the middle point, the fit will be almost exact. (Here, you essentially end up with a Bezier curve, though parametrized a bit differently.)
Given an arbitrary sequence of points in space, how would you produce a smooth continuous interpolation between them?
2D and 3D solutions are welcome. Solutions that produce a list of points at arbitrary granularity and solutions that produce control points for bezier curves are also appreciated.
Also, it would be cool to see an iterative solution that could approximate early sections of the curve as it received the points, so you could draw with it.
The Catmull-Rom spline is guaranteed to pass through all the control points. I find this to be handier than trying to adjust intermediate control points for other types of splines.
This PDF by Christopher Twigg has a nice brief introduction to the mathematics of the spline. The best summary sentence is:
Catmull-Rom splines have C1
continuity, local control, and
interpolation, but do not lie within
the convex hull of their control
points.
Said another way, if the points indicate a sharp bend to the right, the spline will bank left before turning to the right (there's an example picture in that document). The tightness of those turns in controllable, in this case using his tau parameter in the example matrix.
Here is another example with some downloadable DirectX code.
One way is Lagrange polynominal, which is a method for producing a polynominal which will go through all given data points.
During my first year at university, I wrote a little tool to do this in 2D, and you can find it on this page, it is called Lagrange solver. Wikipedia's page also has a sample implementation.
How it works is thus: you have a n-order polynominal, p(x), where n is the number of points you have. It has the form a_n x^n + a_(n-1) x^(n-1) + ...+ a_0, where _ is subscript, ^ is power. You then turn this into a set of simultaneous equations:
p(x_1) = y_1
p(x_2) = y_2
...
p(x_n) = y_n
You convert the above into a augmented matrix, and solve for the coefficients a_0 ... a_n. Then you have a polynomial which goes through all the points, and you can now interpolate between the points.
Note however, this may not suit your purpose as it offers no way to adjust the curvature etc - you are stuck with a single solution that can not be changed.
You should take a look at B-splines. Their advantage over Bezier curves is that each part is only dependent on local points. So moving a point has no effect on parts of the curve that are far away, where "far away" is determined by a parameter of the spline.
The problem with the Langrange polynomial is that adding a point can have extreme effects on seemingly arbitrary parts of the curve; there's no "localness" like described above.
Have you looked at the Unix spline command? Can that be coerced into doing what you want?
There are several algorithms for interpolating (and exrapolating) between an aribtrary (but final) set of points. You should check out numerical recipes, they also include C++ implementations of those algorithms.
Unfortunately the Lagrange or other forms of polynomial interpolation will not work on an arbitrary set of points. They only work on a set where in one dimension e.g. x
xi < xi+1
For an arbitary set of points, e.g. an aeroplane flight path, where each point is a (longitude, latitude) pair, you will be better off simply modelling the aeroplane's journey with current longitude & latitude and velocity. By adjusting the rate at which the aeroplane can turn (its angular velocity) depending on how close it is to the next waypoint, you can achieve a smooth curve.
The resulting curve would not be mathematically significant nor give you bezier control points. However the algorithm would be computationally simple regardless of the number of waypoints and could produce an interpolated list of points at arbitrary granularity. It would also not require you provide the complete set of points up front, you could simply add waypoints to the end of the set as required.
I came up with the same problem and implemented it with some friends the other day. I like to share the example project on github.
https://github.com/johnjohndoe/PathInterpolation
Feel free to fork it.
Google "orthogonal regression".
Whereas least-squares techniques try to minimize vertical distance between the fit line and each f(x), orthogonal regression minimizes the perpendicular distances.
Addendum
In the presence of noisy data, the venerable RANSAC algorithm is worth checking out too.
In the 3D graphics world, NURBS are popular. Further info is easily googled.