Problem
I am writing a ray tracer as a use case for a specific machine learning approach in Computer Graphics.
My problem is that, when I try to find the intersection between a ray and a surface, the result is not exact.
Basically, if I am scattering a ray from point O towards a surface located at (x,y,z), where z = 81, I would expect the solution to be something like S = (x,y,81). The problem is: I get a solution like (x,y,81.000000005).
This is of course a problem, because following operations depend on that solution, and it needs to be the exact one.
Question
My question is: how do people in Computer Graphics deal with this problem? I tried to change my variables from float to double and it does not solve the problem.
Alternative solutions
I tried to use the function std::round(). This can only help in specific situations, but not when the exact solution contains one or more significant digits.
Same for std::ceil() and std::floor().
EDIT
This is how I calculate the intersection with a surface (rectangle) parallel to the xz axes.
First of all, I calculate the distance t between the origin of my Ray and the surface. In case my Ray, in that specific direction, does not hit the surface, t is returned as 0.
class Rectangle_xy: public Hitable {
public:
float x1, x2, y1, y2, z;
...
float intersect(const Ray &r) const { // returns distance, 0 if no hit
float t = (y - r.o.y) / r.d.y; // ray.y = t* dir.y
const float& x = r.o.x + r.d.x * t;
const float& z = r.o.z + r.d.z * t;
if (x < x1 || x > x2 || z < z1 || z > z2 || t < 0) {
t = 0;
return 0;
} else {
return t;
}
....
}
Specifically, given a Ray and the id of an object in the list (that I want to hit):
inline Vec hittingPoint(const Ray &r, int &id) {
float t; // distance to intersection
if (!intersect(r, t, id))
return Vec();
const Vec& x = r.o + r.d * t;// ray intersection point (t calculated in intersect())
return x ;
}
The function intersect() in the previous snippet of code checks for every Rectangle in the List rect if I intersect some object:
inline bool intersect(const Ray &r, float &t, int &id) {
const float& n = NUMBER_OBJ; //Divide allocation of byte of the whole scene, by allocation in byte of one single element
float d;
float inf = t = 1e20;
for (int i = 0; i < n; i++) {
if ((d = rect[i]->intersect(r)) && d < t) { // Distance of hit point
t = d;
id = i;
}
}
// Return the closest intersection, as a bool
return t < inf;
}
The coordinate is then obtained using the geometric interpolation between a line and a surface in the 3D space:
Vec& x = r.o + r.d * t;
where:
r.o: it represents the ray origin. It's defined as a r.o : Vec(float a, float b, float c)
r.d : this is the direction of the ray. As before: r.d: Vec(float d, float e, float f).
t: float representing the distance between the object and the origin.
You could look into using std::numeric_limits<T>::epsilon for your float/double comparison. And see if your result is in the region +-epsilon.
An alternative would be to not ray trace towards a point. Maybe just place relatively small box or sphere there.
Related
I have a triangle, each point of which is defined by a position (X,Y,Z) and a UV coordinate (U,V):
struct Vertex
{
Vector mPos;
Point mUV;
inline Vector& ToVector() {return mPos;}
inline Vector& ToUV() {return mUV;}
};
With this function, I am able to get the UV coordinate at a specific XYZ position:
Point Math3D::TriangleXYZToUV(Vector thePos, Vertex* theTriangle)
{
Vector aTr1=theTriangle->ToVector()-(theTriangle+1)->ToVector();
Vector aTr2=theTriangle->ToVector()-(theTriangle+2)->ToVector();
Vector aF1 = theTriangle->ToVector()-thePos;
Vector aF2 = (theTriangle+1)->ToVector()-thePos;
Vector aF3 = (theTriangle+2)->ToVector()-thePos;
float aA=aTr1.Cross(aTr2).Length();
float aA1=aF2.Cross(aF3).Length()/aA;
float aA2=aF3.Cross(aF1).Length()/aA;
float aA3=aF1.Cross(aF2).Length()/aA;
Point aUV=(theTriangle->ToUV()*aA1)+((theTriangle+1)->ToUV()*aA2)+((theTriangle+2)->ToUV()*aA3);
return aUV;
}
I attempted to reverse-engineer this to make a function that gets the XYZ coordinate from a specific UV position:
Vector Math3D::TriangleUVToXYZ(Point theUV, Vertex* theTriangle)
{
Point aTr1=theTriangle->ToUV()-(theTriangle+1)->ToUV();
Point aTr2=theTriangle->ToUV()-(theTriangle+2)->ToUV();
Point aF1 = theTriangle->ToUV()-theUV;
Point aF2 = (theTriangle+1)->ToUV()-theUV;
Point aF3 = (theTriangle+2)->ToUV()-theUV;
float aA=gMath.Abs(aTr1.Cross(aTr2)); // NOTE: Point::Cross looks like this: const float Cross(const Point &thePoint) const {return mX*thePoint.mY-mY*thePoint.mX;}
float aA1=aF2.Cross(aF3)/aA;
float aA2=aF3.Cross(aF1)/aA;
float aA3=aF1.Cross(aF2)/aA;
Vector aXYZ=(theTriangle->ToVector()*aA1)+((theTriangle+1)->ToVector()*aA2)+((theTriangle+2)->ToVector()*aA3);
return aXYZ;
}
This works MOST of the time. However, it seems to exponentially "approach" the right-angled corner of the triangle-- or something. I'm not really sure what's going on except that the result gets wildly inaccurate the closer it gets to the right-angle.
What do I need to do to this TriangleUVtoXYZ function to make it return accurate results?
I haven't tested your implementation, but you only need to compute two parametric coordinates - the third being redundant since they should sum to 1.
Vector Math3D::TriangleUVToXYZ(Point theUV, Vertex* theTriangle)
{
// T2-T1, T3-T1, P-T1
Point aTr12 = theTriangle[1].ToUV() - theTriangle[0].ToUV();
Point aTr13 = theTriangle[2].ToUV() - theTriangle[0].ToUV();
Point aP1 = theUV - theTriangle[0].ToUV();
// don't need Abs() for the denominator
float aA23 = aTr12.Cross(aTr13);
// parametric coordinates [s,t]
// s = (P-T1)x(T2-T1) / (T3-T1)x(T2-T1)
// t = (P-T1)x(T3-T1) / (T2-T1)x(T3-T1)
float aA12 = aP1.Cross(aTr12) / -aA23;
float aA13 = aP1.Cross(aTr13) / aA23;
// XYZ = V1 + s(V2-V1) + t(V3-V1)
return theTriangle[0].ToVector()
+ aA12 * (theTriangle[1].ToVector() - theTriangle[0].ToVector())
+ aA13 * (theTriangle[2].ToVector() - theTriangle[0].ToVector());
}
I've been trying to implement the Moller-Trumbore ray-triangle intersection algorithm in my raytracing code. The code is supposed to read in a mesh and light sources, fire off rays from the light source, and return the triangle from the mesh which each ray intersects. Here is my implementation of the algorithm:
//Moller-Trumbore intersection algorithm
void getFaceIntersect(modelStruct m, ray r, hitFaceStruct& hitFaces)
{
// Constant thoughout loop
point origin = r.p0;
point direction = r.u;
hitFaces.isHit = false;
for (int i = 0; i < m.faces; i++)
{
// Get face vertices
point v1 = m.vertList[m.faceList[i].v1];
point v2 = m.vertList[m.faceList[i].v2];
point v3 = m.vertList[m.faceList[i].v3];
// Get two edgess
point edge1 = v2 - v1;
point edge2 = v3 - v1;
// Get p
point p = direction.cross(direction, edge2);
// Use p to find determinant
double det = p.dot(edge1, p);
// If the determinant is about 0, the ray lies in the plane of the triangle
if (abs(det) < 0.00000000001)
{
continue;
}
double inverseDet = 1 / det;
point v1ToOrigin = (origin - v1);
double u = v1ToOrigin.dot(v1ToOrigin, p) * inverseDet;
// If u is not between 0 and 1, no hit
if (u < 0 || u > 1)
{
continue;
}
// Used for calculating v
point q = v1ToOrigin.cross(v1ToOrigin, edge1);
double v = direction.dot(direction, q) * inverseDet;
if (v < 0 || (u + v) > 1)
{
continue;
}
double t = q.dot(edge2, q) * inverseDet;
// gets closest face
if (t < abs(hitFaces.s)) {
hitFaceStruct goodStruct = hitFaceStruct();
goodStruct.face = i;
goodStruct.hitPoint = p;
goodStruct.isHit = true;
goodStruct.s = t;
hitFaces = goodStruct;
break;
}
}
}
The relevant code for hitFaceStruct and modelStruct is as follows:
typedef struct _hitFaceStruct
{
int face; // the index of the sphere in question in the list of faces
float s; // the distance from the ray that hit it
bool isHit;
point hitPoint;
} hitFaceStruct;
typedef struct _modelStruct {
char *fileName;
float scale;
float rot_x, rot_y, rot_z;
float x, y, z;
float r_amb, g_amb, b_amb;
float r_dif, g_dif, b_dif;
float r_spec, g_spec, b_spec;
float k_amb, k_dif, k_spec, k_reflective, k_refractive;
float spec_exp, index_refraction;
int verts, faces, norms = 0; // Number of vertices, faces, normals, and spheres in the system
point *vertList, *normList; // Vertex and Normal Lists
faceStruct *faceList; // Face List
} modelStruct;
Whenever I shoot a ray, the values of u or v in the algorithm code always come out to a large negative number, rather than the expected small, positive one. The direction vector of the ray is normalized before I pass it on to the intersection code, and I'm positive I'm firing rays that would normally hit the mesh. Can anyone please help me spot my error here?
Thanks!
I have a set of points that I'm trying to sort in ccw order or cw order from their angle. I want the points to be sorted in a way that they could form a polygon with no splits in its region or intersections. This is difficult because in most cases, it would be a concave polygon.
point centroid;
int main( int argc, char** argv )
{
// I read a set of points into a struct point array: points[n]
// Find centroid
double sx = 0; double sy = 0;
for (int i = 0; i < n; i++)
{
sx += points[i].x;
sy += points[i].y;
}
centroid.x = sx/n;
centroid.y = sy/n;
// sort points using in polar order using centroid as reference
std::qsort(&points, n, sizeof(point), polarOrder);
}
// -1 ccw, 1 cw, 0 collinear
int orientation(point a, point b, point c)
{
double area2 = (b.x-a.x)*(c.y-a.y) - (b.y-a.y)*(c.x-a.x);
if (area2 < 0) return -1;
else if (area2 > 0) return +1;
else return 0;
}
// compare other points relative to polar angle they make with this point
// (where the polar angle is between 0 and 2pi)
int polarOrder(const void *vp1, const void *vp2)
{
point *p1 = (point *)vp1;
point *p2 = (point *)vp2;
// translation
double dx1 = p1->x - centroid.x;
double dy1 = p1->y - centroid.y;
double dx2 = p2->x - centroid.x;
double dy2 = p2->y - centroid.y;
if (dy1 >= 0 && dy2 < 0) { return -1; } // p1 above and p2 below
else if (dy2 >= 0 && dy1 < 0) { return 1; } // p1 below and p2 above
else if (dy1 == 0 && dy2 ==0) { // 3-collinear and horizontal
if (dx1 >= 0 && dx2 < 0) { return -1; }
else if (dx2 >= 0 && dx1 < 0) { return 1; }
else { return 0; }
}
else return -orientation(centroid,*p1,*p2); // both above or below
}
It looks like the points are sorted accurately(pink) until they "cave" in, in which case the algorithm skips over these points then continues.. Can anyone point me into the right direction to sort the points so that they form the polygon I'm looking for?
Raw Point Plot - Blue, Pink Points - Sorted
Point List: http://pastebin.com/N0Wdn2sm (You can ignore the 3rd component, since all these points lie on the same plane.)
The code below (sorry it's C rather than C++) sorts correctly as you wish with atan2.
The problem with your code may be that it attempts to use the included angle between the two vectors being compared. This is doomed to fail. The array is not circular. It has a first and a final element. With respect to the centroid, sorting an array requires a total polar order: a range of angles such that each point corresponds to a unique angle regardless of the other point. The angles are the total polar order, and comparing them as scalars provides the sort comparison function.
In this manner, the algorithm you proposed is guaranteed to produce a star-shaped polyline. It may oscillate wildly between different radii (...which your data do! Is this what you meant by "caved in"? If so, it's a feature of your algorithm and data, not an implementation error), and points corresponding to exactly the same angle might produce edges that coincide (lie directly on top of each other), but the edges won't cross.
I believe that your choice of centroid as the polar origin is sufficient to guarantee that connecting the ends of the polyline generated as above will produce a full star-shaped polygon, however, I don't have a proof.
Result plotted with Excel
Note you can guess from the nearly radial edges where the centroid is! This is the "star shape" I referred to above.
To illustrate this is really a star-shaped polygon, here is a zoom in to the confusing lower left corner:
If you want a polygon that is "nicer" in some sense, you will need a fancier (probably much fancier) algorithm, e.g. the Delaunay triangulation-based ones others have referred to.
#include <stdio.h>
#include <stdlib.h>
#include <math.h>
struct point {
double x, y;
};
void print(FILE *f, struct point *p) {
fprintf(f, "%f,%f\n", p->x, p->y);
}
// Return polar angle of p with respect to origin o
double to_angle(const struct point *p, const struct point *o) {
return atan2(p->y - o->y, p->x - o->x);
}
void find_centroid(struct point *c, struct point *pts, int n_pts) {
double x = 0, y = 0;
for (int i = 0; i < n_pts; i++) {
x += pts[i].x;
y += pts[i].y;
}
c->x = x / n_pts;
c->y = y / n_pts;
}
static struct point centroid[1];
int by_polar_angle(const void *va, const void *vb) {
double theta_a = to_angle(va, centroid);
double theta_b = to_angle(vb, centroid);
return theta_a < theta_b ? -1 : theta_a > theta_b ? 1 : 0;
}
void sort_by_polar_angle(struct point *pts, int n_pts) {
find_centroid(centroid, pts, n_pts);
qsort(pts, n_pts, sizeof pts[0], by_polar_angle);
}
int main(void) {
FILE *f = fopen("data.txt", "r");
if (!f) return 1;
struct point pts[10000];
int n_pts, n_read;
for (n_pts = 0;
(n_read = fscanf(f, "%lf%lf%*f", &pts[n_pts].x, &pts[n_pts].y)) != EOF;
++n_pts)
if (n_read != 2) return 2;
fclose(f);
sort_by_polar_angle(pts, n_pts);
for (int i = 0; i < n_pts; i++)
print(stdout, pts + i);
return 0;
}
Well, first and foremost, I see centroid declared as a local variable in main. Yet inside polarOrder you are also accessing some centroid variable.
Judging by the code you posted, that second centroid is a file-scope variable that you never initialized to any specific value. Hence the meaningless results from your comparison function.
The second strange detail in your code is that you do return -orientation(centroid,*p1,*p2) if both points are above or below. Since orientation returns -1 for CCW and +1 for CW, it should be just return orientation(centroid,*p1,*p2). Why did you feel the need to negate the result of orientation?
Your original points don't appear form a convex polygon, so simply ordering them by angle around a fixed centroid will not necessarily result in a clean polygon. This is a non-trivial problem, you may want to research Delaunay triangulation and/or gift wrapping algorithms, although both would have to be modified because your polygon is concave. The answer here is an interesting example of a modified gift wrapping algorithm for concave polygons. There is also a C++ library called PCL that may do what you need.
But...if you really do want to do a polar sort, your sorting functions seem more complex than necessary. I would sort using atan2 first, then optimize it later once you get the result you want if necessary. Here is an example using lambda functions:
#include <algorithm>
#include <math.h>
#include <vector>
int main()
{
struct point
{
double x;
double y;
};
std::vector< point > points;
point centroid;
// fill in your data...
auto sort_predicate = [¢roid] (const point& a, const point& b) -> bool {
return atan2 (a.x - centroid.x, a.y - centroid.y) <
atan2 (b.x - centroid.x, b.y - centroid.y);
};
std::sort (points.begin(), points.end(), sort_predicate);
}
I'm making a application for school in which I have to click a particular object.
EDIT: This is being made in 2D
I have a rectangle, I rotate this rectangle by X.
The rotation of the rectangle has made my rectangles (x,y,width,height) become a new rectangle around the rotated rectangle.
http://i.stack.imgur.com/MejMA.png
(excuse me for my terrible paint skills)
The Black lines describe the rotated rectangle, the red lines are my new rectangle.
I need to find out if my mouse is within the black rectangle or not. Whatever rotation I do I already have a function for getting the (X,Y) for each corner of the black rectangle.
Now I'm trying to implement this Check if point is within triangle (The same side technique).
So I can either check if my mouse is within each triangle or if theres a way to check if my mouse is in the rotated rectangle that would be even better.
I practically understand everything written in the triangle document, but I simply don't have the math skills to calculate the cross product and the dot product of the 2 cross products.
This is supposed to be the cross product:
a × b = |a| |b| sin(θ) n
|a| is the magnitude (length) of vector a
|b| is the magnitude (length) of vector b
θ is the angle between a and b
n is the unit vector at right angles to both a and b
But how do I calculate the unit vector to both a and b?
And how do I get the magnitude of a vector?
EDIT:
I forgot to ask for the calculation of the dotproduct between 2 cross products.
function SameSide(p1,p2, a,b)
cp1 = CrossProduct(b-a, p1-a)
cp2 = CrossProduct(b-a, p2-a)
if DotProduct(cp1, cp2) >= 0 then return true
else return false
Thank you everyone for your help I think I got the hang of it now, I wish I could accept multiple answers.
If you are having to carry out loads of check, I would shy away from using square root functions: they are computationally expensive. for comparison purposes, just multiply everything by itself and you can bypass the square rooting:
magnitude of vector = length of vector
If vector is defined as float[3] length can be calculated as follows:
double magnitude = sqrt( a[0]*a[0] + a[1]*a[1] + a[2]*a[2] );
However that is expensive computationally so I would use
double magnitudeSquared = a[0]*a[0] + a[1]*a[1] + a[2]*a[2];
Then modify any comparative calculations to use the squared version of the distance or magnitude and it will be more performant.
For the cross product, please forgive me if this maths is shaky, it has been a couple of years since I wrote functions for this (code re-use is great but terrible for remembering things):
double c[3];
c[0] = ( a[1]*b[2] - a[2]*b[1] );
c[1] = ( a[2]*b[0] - a[0]*b[2] );
c[2] = ( a[0]*b[1] - a[1]*b[0] );
To simplify it all I would put a vec3d in a class of its own, with a very simple representation being:
class vec3d
{
public:
float x, y, z;
vec3d crossProduct(vec3d secondVector)
{
vec3d retval;
retval.x = (this.y * secondVector.z)-(secondVector.y * this.z);
retval.y = -(this.x * secondVector.z)+(secondVector.x * this.z);
retval.z = (this.x * secondVector.y)-(this.y * secondVector.x);
return retval;
}
// to get the unit vector divide by a vectors length...
void normalise() // this will make the vector into a 1 unit long variant of itself, or a unit vector
{
if(fabs(x) > 0.0001){
x= x / this.magnitude();
}
if(fabs(y) > 0.0001){
y= y / this.magnitude();
}
if(fabs(z) > 0.0001){
z = / this.magnitude();
}
}
double magnitude()
{
return sqrt((x*x) + (y*y) + (z*z));
}
double magnitudeSquared()
{
return ((x*x) + (y*y) + (z*z));
}
};
A fuller implementation of a vec3d class can be had from one of my old 2nd year coding excercises: .h file and .cpp file.
And here is a minimalist 2d implementation (doing this off the top of my head so forgive the terse code please, and let me know if there are errors):
vec2d.h
#ifndef VEC2D_H
#define VEC2D_H
#include <iostream>
using namespace std;
class Vec2D {
private:
double x, y;
public:
Vec2D(); // default, takes no args
Vec2D(double, double); // user can specify init values
void setX(double);
void setY(double);
double getX() const;
double getY() const;
double getMagnitude() const;
double getMagnitudeSquared() const;
double getMagnitude2() const;
Vec2D normalize() const;
double crossProduct(Vec2D secondVector);
Vec2D crossProduct(Vec2D secondVector);
friend Vec2D operator+(const Vec2D&, const Vec2D&);
friend ostream &operator<<(ostream&, const Vec2D&);
};
double dotProduct(const Vec2D, const Vec2D);
#endif
vec2d.cpp
#include <iostream>
#include <cmath>
using namespace std;
#include "Vec2D.h"
// Constructors
Vec2D::Vec2D() { x = y = 0.0; }
Vec2D::Vec2D(double a, double b) { x = a; y = b; }
// Mutators
void Vec2D::setX(double a) { x = a; }
void Vec2D::setY(double a) { y = a; }
// Accessors
double Vec2D::getX() const { return x; }
double Vec2D::getY() const { return y; }
double Vec2D::getMagnitude() const { return sqrt((x*x) + (y*y)); }
double Vec2D::getMagnitudeSquared() const { return ((x*x) + (y*y)); }
double Vec2D::getMagnitude2 const { return getMagnitudeSquared(); }
double Vec2d::crossProduct(Vec2D secondVector) { return ((this.x * secondVector.getY())-(this.y * secondVector.getX()));}
Vec2D crossProduct(Vec2D secondVector) {return new Vec2D(this.y,-(this.x));}
Vec2D Vec2D::normalize() const { return Vec2D(x/getMagnitude(), y/getMagnitude());}
Vec2D operator+(const Vec2D& a, const Vec2D& b) { return Vec2D(a.x + b.x, a.y + b.y);}
ostream& operator<<(ostream& output, const Vec2D& a) { output << "(" << a.x << ", " << a.y << ")" << endl; return output;}
double dotProduct(const Vec2D a, const Vec2D b) { return a.getX() * b.getX() + a.getY() * b.getY();}
Check if a point is inside a triangle described by three vectors:
float calculateSign(Vec2D v1, Vec2D v2, Vec2D v3)
{
return (v1.getX() - v3.getX()) * (v2.getY() - v3.getY()) - (v2.getX() - v3.getX()) * (v1.getY() - v3.getY());
}
bool isPointInsideTriangle(Vec2D point2d, Vec2D v1, Vec2D v2, Vec2D v3)
{
bool b1, b2, b3;
// the < 0.0f is arbitrary, could have just as easily been > (would have flipped the results but would compare the same)
b1 = calculateSign(point2d, v1, v2) < 0.0f;
b2 = calculateSign(point2d, v2, v3) < 0.0f;
b3 = calculateSign(point2d, v3, v1) < 0.0f;
return ((b1 == b2) && (b2 == b3));
}
In the code above if calculateSign is in the triangle you will get a true returned :)
Hope this helps, let me know if you need more info or a fuller vec3d or 2d class and I can post:)
Addendum
I have added in a small 2d-vector class, to show the differences in the 2d and 3d ones.
The magnitude of a vector is its length. In C++, if you have a vector represented as a double[3], you would calculate the length via
#include <math.h>
double a_length = sqrt( a[0]*a[0] + a[1]*a[1] + a[2]*a[2] );
However, I understand what you actually want is the cross product? In that case, you may want to calculate it directly. The result is a vector, i.e. c = a x b.
You code it like this for example:
double c[3];
c[0] = ( a[2]*b[3] - a[3]*b[2] );
c[1] = ( a[3]*b[1] - a[1]*b[3] );
c[2] = ( a[1]*b[2] - a[2]*b[1] );
You can calculate the magnitude of vector by sqrt(x*x + y*y). Also you can calculate the crossproduct simpler: a x b = a.x * b.y - a.y * b.x. Checking that a point is inside triangle can be done by counting the areas for all 4 triangles. For example a is the area of the source triangle, b,c,d are areas of other ones. If b + c + d = a then the point is inside. Counting the area of triangle is simple: we have vectors a, b that are vertexes of triangle. The area of triangle then is (a x b) / 2
One simple way without getting into vectors is to check for area.
For example ,lets say you have a rectangle with corners A,B,C,D. and point P.
first calculate the area of rectangle, simply find height and width of the rectangle and multiply.
B D
| /
| /
|/____ C
A
For calculating the height,width take one point lets say A, find its distance from all other three points i.e AB,AC,AD 1st and 2nd minimum will be width,and height, max will be diagonal length.
Now store the points from which you get the height, width, lets says those points are B,C.
So now you know how rectangle looks, i.e
B _____ D
| |
|_____|
A C
Then calculate the sum of area of triangles ACP,ABP,BDP,CDP (use heros formula to compute area of rectangle), if it equals to the area of rectangle, point P is inside else outside the rectangle.
I'm trying to create a basic value noise function. I've reached the point where it's outputting it but within the output there are unexpected artefacts popping up such as diagonal discontinuous lines and blurs. I just can't seem to find what's causing it. Could somebody please take a look at it to see if I'm going wrong somewhere.
First off, here are three images that it's ouputting with greater magnification on each one.
//data members
float m_amplitude, m_frequency;
int m_period; //controls the tile size of the noise
vector<vector<float> m_points; //2D array to store the lattice
//The constructor generates the 2D square lattice and populates it.
Noise2D(int period, float frequency, float amplitude)
{
//initialize the lattice to the appropriate NxN size
m_points.resize(m_period);
for (int i = 0; i < m_period; ++i)
m_points[i].resize(m_period);
//populates the lattice with values between 0 and 1
int seed = 209;
srand(seed);
for(int i = 0; i < m_period; i++)
{
for(int j = 0; j < m_period; j++)
{
m_points[i][j] = abs(rand()/(float)RAND_MAX);
}
}
}
//Evaluates a position
float Evaluate(float x, float y)
{
x *= m_frequency;
y *= m_frequency;
//Gets the integer values from each component
int xFloor = (int) x;
int yFloor = (int) y;
//Gets the decimal data in the range of [0:1] for each of the components for interpolation
float tx = x - xFloor;
float ty = y - yFloor;
//Finds the appropriate boundary lattice array indices using the modulus technique to ensure periodic noise.
int xPeriodLower = xFloor % m_period;
int xPeriodUpper;
if(xPeriodLower == m_period - 1)
xPeriodUpper = 0;
else
xPeriodUpper = xPeriodLower + 1;
int yPeriodLower = yFloor % m_period;
int yPeriodUpper;
if(yPeriodLower == m_period - 1)
yPeriodUpper = 0;
else
yPeriodUpper = yPeriodLower + 1;
//The four random values at each boundary. The naming convention for these follow a single 2d coord system 00 for bottom left, 11 for top right
const float& random00 = m_points[xPeriodLower][yPeriodLower];
const float& random10 = m_points[xPeriodUpper][yPeriodLower];
const float& random01 = m_points[xPeriodLower][yPeriodUpper];
const float& random11 = m_points[xPeriodUpper][yPeriodUpper];
//Remap the weighting of each t dimension here if you wish to use an s-curve profile.
float remappedTx = tx;
float remappedTy = ty;
return MyMath::Bilinear<float>(remappedTx, remappedTy, random00, random10, random01, random11) * m_amplitude;
}
Here are the two interpolation functions that it relies on.
template <class T1>
static T1 Bilinear(const T1 &tx, const T1 &ty, const T1 &p00, const T1 &p10, const T1 &p01, const T1 &p11)
{
return Lerp( Lerp(p00,p10,tx),
Lerp(p01,p11,tx),
ty);
}
template <class T1> //linear interpolation aka Mix
static T1 Lerp(const T1 &a, const T1 &b, const T1 &t)
{
return a * (1 - t) + b * t;
}
Some of the artifacts are the result of linear interpolation. Using a higher order interpolation method would help, but it will only solve part of the problem. Crudely put, sharp transitions in the signal can lead to artifacts.
Additional artifacts result from distributing the starting noise values (I.E. the values you are interpolating among) at equal intervals - in this case, a grid. The highest & lowest values will only ever occur at these grid points - at least when using linear interpolation. Roughly speaking, patterns in the signal can lead to artifacts. Two potential ways I know of addressing this part of the problem are either using a nonlinear interpolation &/or randomly nudging the coordinates of the starting noise values to break up their regularity.
Libnoise has an explanation of generating coherent noise which covers these problems & solutions in greater depth with some nice illustrations. You could also peek at the source if you need see how it deals with these problems. And as richard-tingle already mentioned, simplex noise was designed to correct the artifact problems inherent in Perlin noise; it's a little tougher to get your head around, but it's a solid technique.