Points in 3D-Space (d3_point)

Definition

An instance of the data type d3_point is a point in the three-dimensional space R ^3. We use (x,y,z) to denote a point with first (or x-) coordinate x, second (or y-) coordinate y, and third (or z-) coordinate z.

Creation

d3_point	p;	introduces a variable p of type d3_point initialized to the point (0,0,0).
d3_point	p(double x, double y, double z);
		introduces a variable p of type d3_point initialized to the point (x,y,z).
d3_point	p(vector v);	introduces a variable p of type d3_point initialized to the point (v[0],v[1],v[2]). Precondition: v.dim() = 3.

Operations

double	p.xcoord()	returns the first coordinate of p.
double	p.ycoord()	returns the second coordinate of p.
double	p.zcoord()	returns the third coordinate of p.
vector	p.to_vector()	returns the vector $\vec{xyz}$.
point	p.project_xy()	returns p projected into the xy-plane.
point	p.project_yz()	returns p projected into the yz-plane.
point	p.project_xz()	returns p projected into the xz-plane.
double	p.sqr_dist(d3_point q)	returns the square of the Euclidean distance between p and q.
double	p.xdist(d3_point q)	returns the x-distance between p and q.
double	p.ydist(d3_point q)	returns the y-distance between p and q.
double	p.zdist(d3_point q)	returns the z-distance between p and q.
double	p.distance(d3_point q)	returns the Euclidean distance between p and q.
d3_point	p.translate(double dx, double dy, double dz)
		returns p translated by vector (dx,dy,dz).
d3_point	p.translate(vector v)	returns p+v, i.e., p translated by vector v. Precondition: v.dim() = 3.
d3_point	p + vector v	returns p translated by vector v.
d3_point	p - vector v	returns p translated by vector -v.
d3_point	p.reflect(d3_point q, d3_point r, d3_point s)
		returns p reflected across the plane passing through q, r and s.
d3_point	p.reflect(d3_point q)	returns p reflected across point q.
vector	p - q	returns the difference vector of the coordinates.
ostream&	ostream& O << p	writes p to output stream O.
istream&	istream& I >> d3_point& p	reads the coordinates of p (three double numbers) from input stream I.

Non-Member Functions

d3_point	center(d3_point a, d3_point b)
		returns the center of a and b, i.e. $a +\vec{ab}/2$.
d3_point	midpoint(d3_point a, d3_point b)
		returns the center of a and b.
int	orientation(d3_point a, d3_point b, d3_point c, d3_point d)
		computes the orientation of points a, b, c, and d as the sign of the determinant $\left\vert\begin{array}{cccc} a_x & a_y & a_z & 1\\ b_x & b_y & b_z & 1\\ c_x & c_y & c_z & 1\\ d_x & d_y & d_z & 1 \end{array} \right\vert$
double	volume(d3_point a, d3_point b, d3_point c, d3_point d)
		computes the signed volume of the simplex determined by a,b, c, and d, positive if orientation(a,b,c,d) > 0 and negative otherwise.
bool	collinear(d3_point a, d3_point b, d3_point c)
		returns true if points a, b, c are collinear and false otherwise.
bool	coplanar(d3_point a, d3_point b, d3_point c, d3_point d)
		returns true if points a, b, c, d are coplanar and false otherwise.
int	side_of_sphere(d3_point a, d3_point b, d3_point c, d3_point d, d3_point x)
		returns +1 (-1) if point x lies on the positive (negative) side of the oriented sphere through points a, b, c, and d, and 0 if x is contained in this sphere.
int	region_of_sphere(d3_point a, d3_point b, d3_point c, d3_point d, d3_point x)
		determines whether the point x lies inside (= +1), on (= 0), or outside (= -1) the sphere through points a,b,c,d, (equivalent to orientation(a,b,c,d) * side_of_sphere(a,b,c,d,x)) Precondition: orientation(A) != 0
bool	contained_in_simplex(d3_point a, d3_point b, d3_point c, d3_point d, d3_point x)
		determines whether x is contained in the simplex spanned by the points a,b,c,d. Precondition: a,b,c,d are affinely independent.
bool	contained_in_simplex(array<d3_point> A, d3_point x)
		determines whether x is contained in the simplex spanned by the points in A. Precondition: A must have size <= 4 and the points in A must be affinely independent.
bool	contained_in_affine_hull(list<d3_point> L, d3_point x)
		determines whether x is contained in the affine hull of the points in L.
bool	contained_in_affine_hull(array<d3_point> A, d3_point x)
		determines whether x is contained in the affine hull of the points in A.
int	affine_rank(array<d3_point> L)
		computes the affine rank of the points in L.
int	affine_rank(array<d3_point> A)
		computes the affine rank of the points in A.
bool	affinely_independent(list<d3_point> L)
		decides whether the points in A are affinely independent.
bool	affinely_independent(array<d3_point> A)
		decides whether the points in A are affinely independent.
bool	insphere(d3_point a, d3_point b, d3_point c, d3_point d, d3_point e)
		returns true if e lies in the interior of the sphere through a, b, c, and d, and false otherwise.
bool	outsphere(d3_point a, d3_point b, d3_point c, d3_point d, d3_point e)
		returns true if e lies in the exterior of the sphere through a, b, c, and d, and false otherwise.
bool	onsphere(d3_point a, d3_point b, d3_point c, d3_point d, d3_point e)
		returns true if a, b, c, d, and e lie on a common sphere.
d3_point	point_on_positive_side(d3_point a, d3_point b, d3_point c)
		returns point a + (b-a)x (c-a).