Vector Class Reference

Vectors in three dimensions. More...

#include <evector.h>

Public Member Functions
	Vector ()
	Empty.
	Vector (const double &)
	Create a vector with the same coordinates. More...
	Vector (const double &, const double &, const double &)
	Create a vector with argument coordinates. More...
constexpr double &	operator[] (int)
	Gets the i-th coordinate of vector.
constexpr double	operator[] (int) const
	Returns the i-th coordinate of vector.
Vector	operator+ () const
	Overloaded.
Vector	operator- () const
	Overloaded.
Vector &	operator+= (const Vector &)
	Destructive addition.
Vector &	operator-= (const Vector &)
	Destructive subtraction.
Vector &	operator*= (const Vector &)
	Destructively scale a vector by another vector. More...
Vector &	operator/= (const Vector &)
	Destructively divide the components of a vector by another vector.
Vector &	operator*= (const double &)
	Destructive scalar multiply.
Vector &	operator/= (const double &)
	Destructive division by a scalar.
double	Max () const
	Compute the maximum component of a vector. More...
int	MaxIndex () const
	Compute the index of the maximum component of a vector. More...
int	MinIndex () const
	Compute the index of the minimum component of a vector. More...
Vector	Orthogonal () const
	Returns a vector orthogonal to the argument vector. More...
void	RandomOrthonormal (Vector &, Vector &) const
	Computes two random orthonormal vectors to the argument vector. More...
void	Orthonormal (Vector &, Vector &) const
	Given a vector, creates two vectors xand y that form an orthogonal basis. More...
int	Octant (const Vector &) const
	Computes octant index of a vector with respect to the vector object. More...
Vector	Scaled (const Vector &) const
	Scale a vector. More...
Vector	Inverse () const
	Inverse of a vector. More...
Vector	Sort () const
	Sort the terms of the vector into ascending order.
QString	ToString (int=6) const
	Create a QString form a vector. More...
double	Angle (const Vector &) const
	Compute the angle between two vectors. More...
Vector	Fract () const
	Compute the fractional part of the coordinates. More...
Vector	Floor () const
	Compute the numerator part of the coordinates. More...

Static Public Member Functions
static bool	Equal (const Vector &, const Vector &, const double &=0.0001)
	Test if two vectors are almost equal. More...
static Vector	Min (const Vector &, const Vector &)
	Return a vector with coordinates set to the minimum coordinates of the two argument vectors.
static Vector	Max (const Vector &, const Vector &)
	Return a vector with coordinates set to the maximum coordinates of the two argument vectors.
static void	SetMinMax (const Vector &, Vector &, Vector &)
	Update the minimum and maximum values given a vector. More...
static Vector	Mod (const Vector &, const Vector &)
	Modulo of two Vectors. More...
static bool	Coplanar (const Vector &, const Vector &, const Vector &, const double &=1.0e-6)
	Check if three vectors are coplanar. More...
static bool	Coplanar (const Vector &, const Vector &, const Vector &, const Vector &, const double &=1.0e-6)
	Check if four points are coplanar. More...
static Vector	Bilinear (const Vector &, const Vector &, const Vector &, const Vector &, const double &, const double &)
	Bi-linear interpolation between four vectors. More...
static Vector	Trilinear (const Vector &, const Vector &, const Vector &, const Vector &, const Vector &, const Vector &, const Vector &, const Vector &, const double &, const double &, const double &)
	Trilinear interpolation between eight vectors. More...
static double	Slope (const Vector &, const Vector &)
	Compute the vertical slope between two vectors. More...
static Vector	FromString (QString)
	Creates a vector from a QString. More...
static Vector	Polar (const double &, const double &)
	Creates a vector given polar coordinates. More...
static Vector	Solve (const Vector &, const Vector &, const double &, const double &)
	Compute the point on a segment such that the linear function satisfies f(a)=va and f(b)=vb. More...

Static Public Attributes
static const Vector	Null = Vector(0.0, 0.0, 0.0)
	Null vector.
static const Vector	X = Vector(1.0, 0.0, 0.0)
	Vector(1,0,0).
static const Vector	Y = Vector(0.0, 1.0, 0.0)
	Vector(0,1,0).
static const Vector	Z = Vector(0.0, 0.0, 1.0)
	Vector(0,0,1).

Protected Attributes
double	c [3]
	Components.

Friends
bool	operator> (const Vector &, const Vector &)
	Compare two vectors.
bool	operator< (const Vector &, const Vector &)
	Compare two vectors.
bool	operator>= (const Vector &, const Vector &)
	Overloaded.
bool	operator<= (const Vector &, const Vector &)
	Overloaded.
Vector	operator+ (const Vector &, const Vector &)
	Adds up two vectors.
Vector	operator- (const Vector &, const Vector &)
	Difference between two vectors.
constexpr friend double	operator* (const Vector &, const Vector &)
	Scalar product.
Vector	operator* (const Vector &, double)
	Right multiply by a scalar.
Vector	operator* (const double &, const Vector &)
	Left multiply by a scalar.
Vector	operator/ (const Vector &, double)
	Left multiply by a scalar.
Vector	operator/ (const Vector &, const Vector &)
	Cross product.
bool	operator== (const Vector &, const Vector &)
	Strong equality test.
bool	operator!= (const Vector &, const Vector &)
	Strong difference test.
double	Norm (const Vector &)
	Compute the Euclidean norm of a vector. More...
double	SquaredNorm (const Vector &)
	Compute the squared Euclidean norm of a vector. More...
double	NormInfinity (const Vector &)
	Compute the norm infinity of a vector. More...
void	Normalize (Vector &)
	Normalize a vector, computing the inverse of its norm and scaling the components. More...
Vector	Normalized (const Vector &)
	Return a normalized vector. More...
double	Sine (const Vector &, const Vector &)
	Returns the positive sine of two vectors. More...
double	Cosine (const Vector &, const Vector &)
	Returns the positive cosine of two vectors. More...
Vector	Abs (const Vector &)
	Computes the absolute value of a vector. More...
void	Swap (Vector &, Vector &)
	Swap two vectors. More...
void	Swap (Vector *&, Vector *&)
	Swap two pointers to (arrays) vectors. More...
int	Aligned (const Vector &, const Vector &)
	Check if two vectors are aligned. More...
Vector	Clamp (const Vector &, const Vector &, const Vector &)
	Clamp a vector between two bounds. More...
Vector	Lerp (const Vector &, const Vector &, const double &)
	Linear interpolation between two vectors. More...
std::ostream &	operator<< (std::ostream &, const Vector &)
	Overloaded output-stream operator. More...

Detailed Description

Vectors in three dimensions.

Most binary operators have been overloaded as expected, destructive operators, such as addition and subtraction have been implemented and behave as one could expect.

How do I compute the cross product of two vectors?
Simply use the overloaded Vector::operator/, for instance

Vector c=a/b; // Cross product

computes the cross product of a and b.

How do I compute the sine of the angle between two vectors?
Simply use the Sine(const Vector&,const Vector&) function, which internally computes the norm of the cross product divided by the norm of the argument vectors.

double s=Sine(a,b); // Equivalent to Norm(a/b)/(Norm(a)*Norm(b));

How can I get access to the x, y and z components of a vector?
Use v[0], v[1] and v[2] to get access to the x, y and z components of a vector v respectively.

How do I compute the normal of a triangle?
Let a,b,c the vertices of the triangle, simply compute the cross product

Vector n=(a-b)/(a-c);  // Cross product

or use the member function of the Triangle class:

Vector n=Triangle(a,b,c).Normal(); // Compute the normal

How can I sort the three elements in a vector?
Use Vector::Sort() as follows:

Vector s=Vector(2.0,3.0,-1.0).Sort(); // Sort components in ascending order

How do I perform bi-linear interpolation on vectors?
Use Vector::Bilinear() with four vectors and bilinear coordinates. Alternatively, some geometric classes implement bilinear interpolation, such as Quadrangle::Vertex().

Constructor & Destructor Documentation

Vector() [1/2]

Vector::Vector ( const double &  a )

inlineexplicit

Create a vector with the same coordinates.

Parameters

a

Real.

Vector() [2/2]

Vector::Vector ( const double &  a, const double &  b, const double &  c  )

inlineexplicit

Create a vector with argument coordinates.

Parameters

a,b,c

Coordinates.

Member Function Documentation

Angle()

double Vector::Angle

(

const Vector &

b

)

const

Compute the angle between two vectors.

The two input vectors implitly define a frame

Parameters

b	Argument vector forming the angle.

Bilinear()

Vector Vector::Bilinear ( const Vector &  a00, const Vector &  a10, const Vector &  a11, const Vector &  a01, const double &  u, const double &  v  )

inlinestatic

Bi-linear interpolation between four vectors.

The values are given in trigonometric order.

Parameters

a00,a10,a11,a01	Interpolated vectors.
u,v	Interpolation coefficients.

See also

Math::Bilinear

Coplanar() [1/2]

bool Vector::Coplanar ( const Vector &  a, const Vector &  b, const Vector &  c, const double &  epsilon = 1.0e-6  )

static

Check if three vectors are coplanar.

Simply compute the cross product of a and b, and the dot product with c. Compare the result with a given tolerance.

Parameters

a,b,c	Vectors.
epsilon	Tolerance parameter.

Coplanar() [2/2]

bool Vector::Coplanar ( const Vector &  a, const Vector &  b, const Vector &  c, const Vector &  d, const double &  epsilon = 1.0e-6  )

static

Check if four points are coplanar.

Parameters

a,b,c,d	Four points.
epsilon	Tolerance parameter.

Equal()

bool Vector::Equal ( const Vector &  a, const Vector &  b, const double &  epsilon = 0.0001  )

static

Test if two vectors are almost equal.

This function computes the difference between the two argument vectors, then the norm infinity of this difference and check the result againt the epsilon threshold value. This is a convenience function which is the same as:

Vector a,b; // Two vectors
double e; // Epsilon value
bool e=NormInfinity(Abs(b-a))<e?true:false;

Floor()

Vector Vector::Floor ( ) const

inline

Compute the numerator part of the coordinates.

See also

Math::Floor

Fract()

Vector Vector::Fract ( ) const

inline

Compute the fractional part of the coordinates.

See also

Math::Fract

FromString()

Vector Vector::FromString ( QString  n )

static

Creates a vector from a QString.

Parameters

n

String.

Inverse()

Vector Vector::Inverse ( ) const

inline

Inverse of a vector.

This function inverses the components of the vector. This is the same as:

Vector v=Vector(1.0/u[0],1.0/u[1],1.0/u[2]);

Max()

double Vector::Max ( ) const

inline

Compute the maximum component of a vector.

Note that this function is not the same as NormInfinity which computes the maximum of the absolute values of components. The codes are equivalent:

Vector a(-1.0,-3.0,2.0);
double s=NormInfinity(a);
double t=Max(Abs(a));

See also

NormInfinity

MaxIndex()

int Vector::MaxIndex ( ) const

inline

Compute the index of the maximum component of a vector.

Vector a(-1.0,-3.0,2.0);
int i=MaxIndex(Abs(a)); // Should be 1

This function can be used to find the most stretched axis of a bounding box, for instance to cut the box in the middle of this stretched axis:

Box box;
int axis = box.Diagonal().MaxIndex();

See also

Max

MinIndex()

int Vector::MinIndex ( ) const

inline

Compute the index of the minimum component of a vector.

Vector a(-1.0,4.0,2.0);
int i = MinIndex(a); // Should be 0

See also

MaxIndex

Mod()

Vector Vector::Mod ( const Vector &  a, const Vector &  b  )

inlinestatic

Modulo of two Vectors.

Parameters

a,b	Argument vectors.

Octant()

int Vector::Octant

(

const Vector &

p

)

const

Computes octant index of a vector with respect to the vector object.

See also

Box::Octant(const Vector&)

Parameters

p

Vector.

operator*=()

Vector & Vector::operator*= ( const Vector &  u )

inline

Destructively scale a vector by another vector.

This is the same as Scale:

Vector u(2.0,-1.0,1.0);
u=u.Scaled(Vector(3.0,1.0,2.0)); // u*=Vector(3.0,1.0,2.0);

Orthogonal()

Vector Vector::Orthogonal

(

)

const

Returns a vector orthogonal to the argument vector.

The returned orthogonal vector is not computed randomly. First, we find the two coordinates of the argument vector with maximum absolute value. The orthogonal vector is defined by swapping those two coordinates and changing one sign, whereas the third coordinate is set to 0.

The returned orthogonal vector lies in the plane orthogonal to the first vector.

Orthonormal()

void Vector::Orthonormal	(	Vector &	x,
		Vector &	y
	)		const

Given a vector, creates two vectors xand y that form an orthogonal basis.

This algorithm pickes the minor axis in order to reduce numerical instability

Parameters

x,y	Returned vectors such that (x,y,n) form an orthonormal basis (provided n is normalized).

Polar()

Vector Vector::Polar ( const double &  t, const double &  p  )

inlinestatic

Creates a vector given polar coordinates.

Parameters

t	Theta.
p	Phi.

RandomOrthonormal()

void Vector::RandomOrthonormal	(	Vector &	i,
		Vector &	j
	)		const

Computes two random orthonormal vectors to the argument vector.

This function is expensive as it requires the computation of normalized vectors and the evaluation of a random number.

Parameters

i,j	Orthonormal vectors.

Scaled()

Vector Vector::Scaled ( const Vector &  a ) const

inline

Scale a vector.

Parameters

a	Scaling vector.

SetMinMax()

void Vector::SetMinMax ( const Vector &  x, Vector &  a, Vector &  b  )

static

Update the minimum and maximum values given a vector.

Parameters

x	Input vector.
a,b	Lower and upper vectors that will be updated.

Slope()

double Vector::Slope ( const Vector &  a, const Vector &  b  )

inlinestatic

Compute the vertical slope between two vectors.

This is a convenience function.

Parameters

a,b	Argument vectors.

Solve()

Vector Vector::Solve ( const Vector &  a, const Vector &  b, const double &  va, const double &  vb  )

inlinestatic

Compute the point on a segment such that the linear function satisfies f(a)=va and f(b)=vb.

This function can be used as a first approximation for computing the intersection between a segment and an implicit surface.

See also

Linear::Solve(const double& a, const double& b, const Vector&, const Vector&);

ToString()

QString Vector::ToString

(

int

n = 6

)

const

Create a QString form a vector.

The implementation currently uses:

return QString("Vector(%1,%2,%3)").arg(c[0],0,'f',n).arg(c[1],0,'f',n).arg(c[2],0,'f',n);

Another possible implementation could have been:

return "Vector("+QString::number(c[0],n)+","+QString::number(c[1],n)+","+QString::number(c[2],n)+")";

Parameters

n	Number of digits.

Trilinear()

Vector Vector::Trilinear ( const Vector &  a, const Vector &  b, const Vector &  c, const Vector &  d, const Vector &  e, const Vector &  f, const Vector &  g, const Vector &  h, const double &  u, const double &  v, const double &  w  )

static

Trilinear interpolation between eight vectors.

See also

Math::Bilinear()

Parameters

a,b,c,d,e,f,g,h	Interpolated values.
u,v,w	Interpolation coefficients.

Friends And Related Function Documentation

Abs

Vector Abs ( const Vector &  u )

friend

Computes the absolute value of a vector.

Parameters

u

Vector.

Aligned

int Aligned ( const Vector &  u, const Vector &  v  )

friend

Check if two vectors are aligned.

Computes the cosine of the two vectors, and checks for unity.

Parameters

u,v

Vectors.

Clamp

Vector Clamp ( const Vector &  x, const Vector &  a, const Vector &  b  )

friend

Clamp a vector between two bounds.

Parameters

x	Input vector
a,b	Vector bounds.

Cosine

double Cosine ( const Vector &  u, const Vector &  v  )

friend

Returns the positive cosine of two vectors.

Basically computes the dot product of the normalized vectors.

Parameters

u,v

Vectors.

Lerp

Vector Lerp ( const Vector &  a, const Vector &  b, const double &  t  )

friend

Linear interpolation between two vectors.

Parameters

a,b	Interpolated points.
t	Interpolant.

Norm

double Norm ( const Vector &  u )

friend

Compute the Euclidean norm of a vector.

This function involves a square root computation, it is in general more efficient to rely on the squared norm of a vector instead.

Parameters

u

Vector.

See also

SquaredNorm

Normalize

void Normalize ( Vector &  u )

friend

Normalize a vector, computing the inverse of its norm and scaling the components.

This function does not check if the vector is null, which might resulting in errors.

Normalized

Vector Normalized ( const Vector &  u )

friend

Return a normalized vector.

Compute the inverse of its norm and scale the components.

This function does not check if the vector is null.

Parameters

u

Vector.

NormInfinity

double NormInfinity ( const Vector &  u )

friend

Compute the norm infinity of a vector.

See also

Max

Parameters

u

Vector.

operator<<

std::ostream & operator<< ( std::ostream &  s, const Vector &  u  )

friend

Overloaded output-stream operator.

Parameters

u	Vector.
s	Stream.

Sine

double Sine ( const Vector &  u, const Vector &  v  )

friend

Returns the positive sine of two vectors.

Computes the cross product of the vectors and normalizes the result.

Parameters

u,v

Vectors.

SquaredNorm

double SquaredNorm ( const Vector &  u )

friend

Compute the squared Euclidean norm of a vector.

Parameters

u

Vector.

See also

Norm

Swap [1/2]

void Swap ( Vector &  a, Vector &  b  )

friend

Swap two vectors.

Parameters

a,b

Vectors.

Swap [2/2]

void Swap ( Vector *&  a, Vector *&  b  )

friend

Swap two pointers to (arrays) vectors.

Parameters

a,b

Vectors.