I'm currently making a prototype game where the player walks along the isosurface of a collection of moving metaballs. I've already implemented walking along a sphere with a forward and up vector. To extend this to the metaballs I need to be able to query for an arbitrary point the direction to the closest surface and the distance to the closest surface to be able to snap the player back to the surface after I've moved him along the forward vector. I can calculate the direction by taking a weighted average of all vectors but how do I get the distance?
I'm using the 1 / (x*x + y*y + z*z) function with an isosurface of 1 for my metaballs but I would appreciate any generalization so that I can use the same approach for other shapes.
In general, you would just derive your function to get the normal on the surface. Blackpawn has a nice explanation how to do it with your specific case here.
Once you got the normal, move along its direction until you hit the isosurface (this is generic "root" finding).
Related
I struggle with the orientation of an object I am moving along a hermite curve.
I figured out how to move it at constant speed at also have the tangent of my curve, which would be the forward vector of the moving object. My problem is: How do I know the up and right vector? The easiest way would be to start at a given rotation and then step through the curve always taking the last rotation as a reference for the next one, like in this reference:
Camera movement along a splinepdf
But this would result in an uncontrollable rotation at the end of the spline. What I am trying to do is to create an algorithm which gives you the correct orientation at any point of the curve, without stepping through it. Ideally it would use the orientation of the two controlpoints for the current segment as a reference.
I thought of using some kind of pre-calculated data, which is created from the two orientations of the controlpoints and the current curve segments form, but didn't manage to come up with a solution.
I would be happy to get any answers or just ideas how to approach this problem.
Let C(t) be the camera trajectory, with tangent vector T(t). The tangent vector controls the pitch and the yaw. What you are missing is roll control.
Define an auxiliary trajectory D(t) that "parallels" C(t) and use the vector CD(t). The up vector is given by U(t)=T(t) /\ CD(t) (normalized), and the right vector by U(t) /\ T(t) (normalized).
OK i came up with a solution using frenet frames. I define an orientation for each of my control points, then i calculate a number of points along the spline for each segment. Each points orientation is then calculated using the previous points orientation. The orientatin of the first point equals the orientation of the control point.
Here is a very nice description of the procedure.
After calculating each points orientation, you can interpolate them so the last points orientation matches the orientation of the next controlpoint.
I'm trying to take an array of 3D points and a plane and divide the points up into 2 arrays based on which side of the plane they are on. Before I get to heavily into debugging I wanted to post what I'm planning on doing to make sure my understanding of how to do this will work.
Basically I have the plane with 3 points and I use (pseudo code):
var v1 = new vector(plane.b.x-plane.a.x, plane.b.y-plane.a.y, plane.b.z-plane.a.z);
var v2 = new vector(plane.c.x-plane.a.x, plane.c.y-plane.a.y, plane.c.z-plane.a.z);
I take the cross product of these two vectors to get the normal vector.
Then I loop through my array of points and turn them into vectors and calculate the dot product against the normal.
Then i use the dot product to determine the side that the point is on.
Does this sound like it would work?
Let a*x+b*y+c*z+d=0 be the equation determining your plane.
Substitute the [x,y,z] coordinates of a point into the left hand side of the equation (I mean the a*x+b*y+c*z+d) and look at the sign of the result.
The points having the same sign are on the same side of the plane.
Honestly, I did not examine the details of what you wrote. I guess you agree that what I propose is simpler.
Following the 'put points into the plane's equation and check the sign' approach given previously. The equation can be easily obtained using SymPy. I used it to find location of points (saved as numpy arrays) in a list of points.
from sympy import Point3D, Plane
plane=Plane(Point3D(point1), Point3D(point2), Point3D(point3))
for point in pointList:
if plane.equation(x=point[0], y=point[1],z=point[2]) > 0:
print "point is on side A"
else:
print "point is on side B"
I haven't tested its speed compared to other methods mentioned above but is definitely the easiest method.
Your approach sounds good. However, when you say "and turn them into vectors", it might not be good (depending on the meaning of your sentence).
You should "turn your points into vector" by computing the difference in terms of coordinates between the current point and one of the points in the plane (for example, one of the 3 points defining the plane). As you wrote it, it sounds like you might have misunderstood that ; but apart from that, it's ok!
take into account the normal vector of the plane
example: for the point A=[-243.815437431962, -41.7407630281635, 10.0]
equation= -2663.1860000000006*Z +21305.488000000005=0
RESULt POSITIVE
but if equation= 2663.1860000000006*Z -21305.488000000005=0
RESULT NEGATIVE
I am taking Computer Graphics course.In 3D,I have a point and a polygon and I want to determine this point is located above or below my polygon.Thanks for your replies,in advance.
If above or below the plane on which the polygon is resting will do, you can compare the dot product of the point onto the plane normal and that of any point on the plane. Or look at the sign of the dot product between the normal and a vector from a point on the plane to the point, if you prefer.
To check whether it is actually 'above' or 'below' in the sense of being directly above or below (ie, not off to the side somewhere) then do a point in polygon by projecting the whole thing into 2d along the normal and then a distance along normal test.
It depends on your definition of above and below, let me first talk about the easy case:
If you think of above/below in terms of a global direction (typically the y-axis or z-axis), just compare the values on that axis.
Ok, now the more difficult interpretation: On which side of the polygon is the point.
Unless it is complanar you cannot decide it for the polygon at once. So if it is non-complanar you have to tesselate it into triangles and decide for each of them.
For a triangle you can decide whether a point is above or below it (in 3D), first calculate the cross product of 2 vectors that make up the sides of the triangle; this will define a direction (= the definition of "above" and "below"), this depends on the order in which you use those 2 vectors so be careful. Then calculate the dot-product of the new vector (which is called the perpendicular of that triangle) and the difference-vector of the point-to-test and the triangle-base.
I am trying to animate an object, let's say its a car. I want it go from point
x1,y1,z1
to point x2,y2,z2 . It moves to those points, but it appears to be drifting rather than pointing in the direction of motion. So my question is: how can I solve this issue in my updateframe() event? Could you point me in the direction of some good resources?
Thanks.
First off how do you represent the road?
I recently done exactly this thing and I used Catmull-Rom splines for the road. To orient an object and make it follow the spline path you need to interpolate the current x,y,z position from a t that walks along the spline, then orient it along the Frenet Coordinates System or Frenet Frame for that particular position.
Basically for each point you need 3 vectors: the Tangent, the Normal, and the Binormal. The Tangent will be the actual direction you will like your object (car) to point at.
I choose Catmull-Rom because they are easy to deduct the tangents at any point - just make the (vector) difference between 2 other near points to the current one. (Say you are at t, pick t-epsilon and t+epsilon - with epsilon being a small enough constant).
For the other 2 vectors, you can use this iterative method - that is you start with a known set of vectors on one end, and you work a new set based on the previous one each updateframe() ).
You need to work out the initial orientation of the car, and the final orientation of the car at its destination, then interpolate between them to determine the orientation in between for the current timestep.
This article describes the mathematics behind doing the interpolation, as well as some other things to do with rotating objects that may be of use to you. gamasutra.com in general is an excellent resource for this sort of thing.
I think interpolating is giving the drift you are seeing.
You need to model the way steering works .. your update function should 1) move the car always in the direction of where it is pointing and 2) turn the car toward the current target .. one should not affect the other so that the turning will happen and complete more rapidly than the arriving.
In general terms, the direction the car is pointing is along its velocity vector, which is the first derivative of its position vector.
For example, if the car is going in a circle (of radius r) around the origin every n seconds then the x component of the car's position is given by:
x = r.sin(2πt/n)
and the x component of its velocity vector will be:
vx = dx/dt = r.(2π/n)cos(2πt/n)
Do this for all of the x, y and z components, normalize the resulting vector and you have your direction.
Always pointing the car toward the destination point is simple and cheap, but it won't work if the car is following a curved path. In which case you need to point the car along the tangent line at its current location (see other answers, above).
going from one position to another gives an object a velocity, a velocity is a vector, and normalising that vector will give you the direction vector of the motion that you can plug into a "look at" matrix, do the cross of the up with this vector to get the side and hey presto you have a full matrix for the direction control of the object in motion.
I have a bunch of latitude/longitude pairs that map to known x/y coordinates on a (geographically distorted) map.
Then I have one more latitude/longitude pair. I want to plot it on the map as best is possible. How do I go about doing this?
At first I decided to create a system of linear equations for the three nearest lat/long points and compute a transformation from these, but this doesn't work well at all. Since that's a linear system, I can't use more nearby points either.
You can't assume North is up: all you have is the existing lat/long->x/y mappings.
EDIT: it's not a Mercator projection, or anything like that. It's arbitrarily distorted for readability (think subway map). I want to use only the nearest 5 to 10 mappings so that distortion on other parts of the map doesn't affect the mapping I'm trying to compute.
Further, the entire map is in a very small geographical area so there's no need to worry about the globe--flat-earth assumptions are good enough.
Are there any more specific details on the kind of distortion? If, for example, your latitudes and longitudes are "distorted" onto your 2D map using a Mercator projection, the conversion math is readily available.
If the map is distorted truly arbitrarily, there are lots of things you could try, but the simplest would probably be to compute a weighted average from your existing point mappings. Your weights could be the squared inverse of the x/y distance from your new point to each of your existing points.
Some pseudocode:
estimate-latitude-longitude (x, y)
numerator-latitude := 0
numerator-longitude := 0
denominator := 0
for each point,
deltaX := x - point.x
deltaY := y - point.y
distSq := deltaX * deltaX + deltaY * deltaY
weight := 1 / distSq
numerator-latitude += weight * point.latitude
numerator-longitude += weight * point.longitude
denominator += weight
return (numerator-latitude / denominator, numerator-longitude / denominator)
This code will give a relatively simple approximation. If you can be more precise about the way the projection distorts the geographical coordinates, you can probably do much better.
Alright. From a theoretical point of view, given that the distortion is "arbitrary", and any solution requires you to model this arbitrary distortion, you obviously can't get an "answer". However, any solution is going to involve imposing (usually implicitly) some model of the distortion that may or may not reflect the reality of the situation.
Since you seem to be most interested in models that presume some sort of local continuity of the distortion mapping, the most obvious choice is the one you've already tried: linear interpolaton between the nearest points. Going beyond that is going to require more sophisticated mathematical and numerical analysis knowledge.
You are incorrect, however, in presuming you cannot expand this to more points. You can by using a least-squared error approach. Find the linear answer that minimizes the error of the other points. This is probably the most straight-forward extension. In other words, take the 5 nearest points and try to come up with a linear approximation that minimizes the error of those points. And use that. I would try this next.
If that doesn't work, then the assumption of linearity over the area of N points is broken. At that point you'll need to upgrade to either a quadratic or cubic model. The math is going to get hectic at that point.
the problem is that the sphere can be distorted a number of ways, and having all those points known on the equator, lets say, wont help you map points further away.
You need better 'close' points, then you can assume these three points are on a plane with the fourth and do the interpolation --knowing that the distance of longitudes is a function, not a constant.
Ummm. Maybe I am missing something about the question here, but if you have long/lat info, you also have the direction of north?
It seems you need to map geodesic coordinates to a projected coordinates system. For example osgb to wgs84.
The maths involved is non-trivial, but the code comes out a only a few lines. If I had more time I'd post more but I need a shower so I will be boring and link to the wikipedia entry which is pretty good.
Note: Post shower edited.