Calculate bounce angle aggainst rotated line - c++

I have been trying to implement The second answer of this question
My variables are mainly named the same as in the link.
Here they calculate the bounce angle of a point on a rotated surface with normals,
But I can't seem to do it, am I doing something wrong?
The value output I'm expecting is the new velocity after bouncing on the rotated line, currently my output is doing all crazy kind of things, bouncing to high, bouncing the other way and sometimes the right way but mostly ignoring the rotated angle of my line.
Here's my current Code:
Variables:
Top is the first point of my line
Right is the second point of my line
n is the normal
v is the velocity
dotv is n * v
dotn is n * n
Old Code:
sf::Vector2f n(-(top.y - right.y),(top.x - right.x));
sf::Vector2f dotv = sf::Vector2f(ball.getSpeed().x * n.x, ball.getSpeed().y * n.y);
sf::Vector2f dotn = sf::Vector2f(n.x*n.x,n.y*n.y);
sf::Vector2f u = sf::Vector2f(dotv.x/dotn.x,dotv.y/dotn.y);
u = sf::Vector2f(u.x*n.x,u.y*n.y);
sf::Vector2f w = ball.getSpeed() - u;
ball.setSpeed(sf::Vector2f((sf::Vector2f(w.x*0.5,w.y*0.5)-u)));
Edit:
New Code:
sf::Vector2f n(-(top.y - right.y),(top.x - right.x));
double dotv = ball.getSpeed().x*n.x + ball.getSpeed().y*n.y;
double dotn = n.x*n.x + n.y*n.y;
sf::Vector2f u = sf::Vector2f((dotv/dotn)*n.x, (dotv/dotn)*n.y);
sf::Vector2f w = ball.getSpeed() - u;
ball.setSpeed(sf::Vector2f(ball.getSpeed().x,ball.getSpeed().y) - w);
At first I made the mistake of calculating dotproducts as vectors this has been resolved, now it still gives me a strange output It fires my ball directly trough my object in the angle of the reversed normal
Any help would be greatly appreciated.

One problem I see is that you have dot products as vectors. A dot product results in a scalar (single value).
Also, to make your life a lot easier, you make functions for vector operations and even overload operators when appropriate. E.g. vector addition, subtraction. It's probably best to regular functions for dot product and cross product. Here are some examples of what I mean
class Vector2f
{
// Member of Vector2f
X& operator+=(const Vector2f& other)
{
x += other.x;
y += other.y;
return *this;
}
};
// Not member of Vector2f
inline Vector2f operator+(Vector2f a, const Vector2f& b)
{
a += b;
return a;
}
double dot(const Vector2f& a, const Vector2f& b)
{
return a.getX()*b.getX() + a.getY()*b.getY();
}

Related

2d Elastic Collision with Circles

I've seen there is a lot of posts about this already but I can't find one that relates to what I want to do,
I used the formula from here:
https://www.vobarian.com/collisions/2dcollisions2.pdf
As well as this one:
https://www.plasmaphysics.org.uk/programs/coll2d_cpp.htm
I think they area basically the same thing, now my problem is one of my circles is always static, and what I want is when the other circle hits it straight on, I want it to bounce back with the same speed, but these formulas have the circle stop still, presumably as it would pass it's energy to the other circle which would then move away.
I tried doing things like bounce = vel.x pre collision - vel.y post collision and add or subtract that to vel.x post collision and it kinda works but not really, the angles are wrong and depending on which direction the ball is coming from it may bounce up instead of down, left instead of right,
would probably require a lot of if/else statements to get to work at all.
Can someone suggest something?
here's the code for the function :
void Collision2(sf::CircleShape* b1, sf::CircleShape* b2, sf::Vector2f vel1,sf::Vector2f& vel2) {
//vel1 is 0,0 but i might want to use it later
//mass
float m1 = 10;
float m2 = 10;
//normal vector
sf::Vector2f nVec((b2->getPosition().x - b1->getPosition().x), (b2->getPosition().y - b1->getPosition().y));
//unit vector
sf::Vector2f uNVec(nVec / sqrt((nVec.x * nVec.x) + (nVec.y * nVec.y)));
//unit tangent vec
sf::Vector2f uTVec(-uNVec.y, uNVec.x);
float v1n = (uNVec.x * vel1.x) + (uNVec.y * vel1.y);
float v2n = (uNVec.x * vel2.x) + (uNVec.y * vel2.y);
float v1t = uTVec.x * vel1.x + uTVec.y * vel2.y;
float v2t = (uTVec.x * vel2.x) + (uTVec.y * vel2.y);
//v1t and v1n after collision
float v1tN = v1t;
float v2tN = v2t;
float v1nN = (v1n * (m1 - m2) + (2 * m2) * v2n) / (m1 + m2);
float v2nN = (v2n * (m2 - m1) + (2 * m1) * v1n) / (m1 + m2);
//new velocities
sf::Vector2f vel1N(v1nN*uNVec);
sf::Vector2f vel1tN(v1tN * uTVec);
sf::Vector2f vel2N(v2nN * uNVec);
sf::Vector2f vel2tN(v2tN * uTVec);
vel1 = (vel1N + vel1tN);
vel2 = (vel2N + vel2tN);
}
Physics part
The sources you added illustrate the physics behind it very well. when the two balls collide they transfer momentum between them. In an elastic collision this transfer keeps the energy of the system, the same.
We can think of the collision in terms of inertia and momentum, rather than starting from velocity. The kinetic energy of a body is generally p^2/(2m), so if we transfer dp from the moving body then we will have change in energy: dE = -pdp/m + dp^2/(2m) + dp^2/(2M) = 0. Here m is the moving and M is the stationary mass. Rearranging gives pdp/m = dp^2*(1/(2m) + 1/(2M)). We can consider m = M yielding p = dp (i.e. All moment is transferred (Note: this is a simplistic view, only dealing with head on collisions)). In the limit where the stationary object is massive however (M >> m) the result will be dp = 2p, simply bouncing off.
Programming
You can achieve the results by setting M to the maximum allowed float value (if I recall 1/inf == NaN in the IEEE standard so that doesn't work unfortunately). Alternatively you can do the collision within the circle by creating custom classes like:
class Circle : public sf::CircleShape{
public:
virtual collide (Circle*);
}
class StaticCircle : public Circle{
public:
collide (Circle*) override;
}
in the second one you can omit any terms where you divide by the mass of the circle, as it is in essence infinite.

C++ - Deal with floating point errors in geometric interpolation

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.

Calculating velocity from speed and direction

I have an object which wants to move from point A to point B, at a certain speed (using SFML C++).
sf::Vector2f aPos; //sf::Vector2f has two members x and y, stored as floats
sf::Vector2f bPos;
float speed; //pixels per tick
I want to use these variables to find out a velocity, with an x speed and a y speed, such that the hypotenuse of vel.x and vel.y has length speed, and has the same bearing as that from point aPos to point bPos.
sf::Vector2f vel;
What is the simplest and most effective function that I can write, which takes sf::Vector2f aPos, sf::Vector2f bPos and float speed as arguments, and returns sf::Vector2f vel
sf::Vector2f findVel(sf::Vector2f aPos, sf::Vector2f bPos, float speed) {
sf::Vector2f vel;
//the code I need
return vel;
}
Thank you in advance! :)
Taking k_g's answer, we can actually shorten it quite a bit given that Vector2f has operator overloads. It can simply become:
sf::Vector2f findVel(const sf::Vector2f& aPos, const sf::Vector2f& bPos, float speed) {
sf::Vector2f disp = bPos-aPos;
float distance = sqrt(disp.x*disp.x+disp.y*disp.y); // std::hypot(disp.x, disp.y) if C++ 11
return disp * (speed/distance);
}
This also allows us to benefit from any optimizations that might have been built into the Vector class for operations.
Note the change to const & for the input parameters; no need to make a copy of them since you aren't manipulating them.
well, we know that speed == distance/time and velocity == displacement/time.
Solving for 1/time in the first equation, we get 1/time==speed/distance. We substitute this into the second equation to get velocity == displacement*speed/distance.
Using this formula, we get the following function
sf::Vector2f findVel(sf::Vector2f aPos, sf::Vector2f bPos, float speed) {
sf::Vector2f vel;
float dispx = bPos.x-aPos.x;
float dispy = bPos.y-aPos.y;
float distance = sqrt(dispx*dispx+dispy*dispy);
vel.x = dispx*speed/distance;
vel.y = dispy*speed/distance;
return vel;
}

Ray tracing - refraction bug

I am writing a ray tracer. So far I have diffuse, Blinn lighting and reflections. Something has gone wrong with my refractions and I have no idea what. I'm hoping someone can help me out.
I have a big red diffuse + Blinn sphere and a small refractive one with refraction index n = 1.5.
The small one is just really screwed up.
Relevant code:
ReflectiveSurface::ReflectiveSurface(const Color& _n, const Color& _k) :
F0(Color(((_n - 1)*(_n - 1) + _k * _k) / ((_n + 1)*(_n + 1) + _k * _k))) {}
Color ReflectiveSurface::F(const Point& N, const Point& V) const {
float cosa = fabs(N * V);
return F0 + (F0 * (-1) + 1) * pow(1 - cosa, 5);
}
Color ReflectiveSurface::getColor(const Incidence& incidence, const Scene& scene, int traceDepth) const {
Point reflectedDir = reflect(incidence.normal, incidence.direction);
Ray ray = Ray(incidence.point + reflectedDir * epsilon, reflectedDir);
return F(incidence.normal, incidence.direction) * scene.rayTrace(ray, traceDepth + 1);
}
Point ReflectiveSurface::reflect(const Point& N, const Point& V) const {
return V - N * (2 * (N * V));
}
bool RefractiveSurface::refractionDir(Point& T, Point& N, const Point& V) const {
float cosa = -(N * V), cn = n;
if (cosa < 0) { cosa = -cosa; N = N * (-1); cn = 1 / n; }
float disc = 1 - (1 - cosa * cosa) / cn / cn;
if (disc < 0) return false;
T = V / cn + N * (cosa / cn - sqrt(disc));
return true;
}
RefractiveSurface::RefractiveSurface(float _n, const Color& _k) : ReflectiveSurface(Color(1, 1, 1) * _n, _k) {}
Surface* RefractiveSurface::copy() { return new RefractiveSurface(*this); }
Color RefractiveSurface::getColor(const Incidence& incidence, const Scene& scene, int traceDepth) const {
Incidence I = Incidence(incidence);
Color reflectedColor, refractedColor;
Point direction = reflect(I.normal, I.direction);
Ray reflectedRay = Ray(I.point + direction * epsilon, direction);
if (refractionDir(direction, I.normal, I.direction)) {
Ray refractedRay = Ray(I.point + direction * epsilon, direction);
Color colorF = F(I.normal, I.direction);
reflectedColor = colorF * scene.rayTrace(reflectedRay, traceDepth + 1);
refractedColor = (Color(1, 1, 1) - colorF) * scene.rayTrace(refractedRay, traceDepth + 1);
}
else {
reflectedColor = scene.rayTrace(reflectedRay, traceDepth + 1);
}
return reflectedColor + refractedColor;
}
The code is all over the place, since this is a homework and I'm not allowed to include additional headers and I have to send it in in one cpp file, so i had to separate every class into forward declaration, declaration and implementation in that one file. It makes me vomit but I tried to keep it as clean as possible. There is tons of code so I only included what I thought was most related. ReflectiveSurface is RefractiveSurface's parent class. N is the surface normal, V is the ray direction vector this normal, n is the refraction index. The incidence structure holds a point, a normal and a direction vector.
Formulas for the Fersnel approximation and the refraction vector respectively:
You can see in the code that I use an epsilon * ray direction value to avoid shadow acne caused by float imprecision. Something similar seems to be happening to the small sphere, though.
Another screenshot:
As you can see, the sphere doesn't appear transparent, but it does inherit the diffuse sphere's color. It also usually has some white pixels.
Without refraction:
RefractiveSurface::refractionDir takes the normal N by (non-const) reference, and it may invert it. This seems dangerous. It's not clear the caller wants I.normal to be flipped, as it's used in color calculations further down.
Also, refracted_color is not always initialized (unless the Color constructor makes it black).
Try (temporarily) simplifying and just see if the refracted rays hit where you expect. Remove the Fresnel computation and the reflection component and just set refracted_color to the result of the trace of the refracted ray. That will help determine if the bug is in the Fresnel calculation or in the geometry of bending the ray.
A debugging tip: Color the pixels that don't hit anything with something other than black. That makes it easy to distinguish the misses from the shadows (surface acne).
The answer turned out to be pretty simple, but it took me like 3 days of staring at the code to catch the bug. I have a Surface class, I derive from it two classes: RoughSurface (diffuse+blinn) and RelfectiveSurface. Then, RefractiveSurace is derived from RefleciveSurface. ReflectiveSurface's constructor takes the refractive index(n) and the extinction value (k) as parameters, but doesn't store them. (F0) is computed from them during construction, and then they are lost. RefractiveSurface, on the other hand, uses (n) in the refraction angle calculation.
Old constructor:
RefractiveSurface::RefractiveSurface(float _n, const Color& _k) :
ReflectiveSurface(Color(1, 1, 1) * _n, _k) {}
New Constructor:
RefractiveSurface::RefractiveSurface(float _n, const Color& _k) :
ReflectiveSurface(Color(1, 1, 1) * _n, _k), n(_n) {}
As you can see, I forgot to save the (n) value for RefractiveSurface in the constructor.
Small red sphere behind big glass sphere lit from the two sides of the camera:
It looks awesome in motion!D
Thank you for your time, guys. Gotta finish this homework, then I'll rewrite the whole thing and optimize the hell out of it.

Check if mouse is within triangle C++

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.