householder

Arguments

v

real or complex column vector

w

real or complex column vector with same size as v. Default value is eye(v) ((Ox) axis).

u

unit vector lying in the (v,w) plane and orthogonal to the bisectrix of (v,w). Column of size(v) of real or complex numbers.

H

Orthogonal Householder reflexion matrix: H= eye() - 2*u*u'. H is such that inv(H)==H, H'==H, and det(H)==-1.

If v and w are real, H*v is proportional to w.

Description

householder(..) computes the unit vector u lying in the (v,w) plane and orthogonal to the bisectrix of (v,w).

If v and w are proportional:

• If they are opposite, u= v/|v| is returned.

• If they are real and have the same direction, u is set in the (xOy) plane with a priori u(1)>0, and orthogonal to v (u'*v==0). However,
  • If they are along (Ox), u = (Oy+) is returned instead.
  • If v and w are scalars with same signs, the orthogonal sub-space is restricted to {0} that can't be normalized: u and H are then set to %nan.

If the related reflexion matrix H is computed, for any point A of column coordinates a, H*a are the coordinates of the symetrical image of A with respect to the (v,w) plane (see the example below).

If v or/and w are in row, they are priorly transposed into columns.

If v or/and w are [], [] is returned for u and H.

Examples

a = [ rand(1,1) 0  0 ]';
[ra hm] = householder(a);
[a ra hm*a ]
norm(ra)

b = rand(3,1);
[rb, hm] = householder(b);
[b rb eye(b) clean(hm*b) ]
norm(rb)

[rb2b, hm] = householder(b, 2*b);
[b rb2b clean(hm*b ./ b) ]  // last column must be uniform
norm(rb2b)                  // must be 1

c = rand(3,1);
[rbc, hm] = householder(b,c);
norm(rbc)          // must be 1
hm*b ./c           // must be uniform

d = b + %i*c;
e = rand(3,1) + %i*rand(3,1);
[rde, hm] = householder(d,e);
norm(rbc)               // must be 1
clean(inv(hm) - hm)     // must be zeros(3,3)
clean(hm' - hm)         // must be zeros(3,3)
clean(det(hm))          // must be -1

Application: Symetrical image of an object w.r.t. a given plane

// (OA) = [0 0 1] is reflected in O into (OB) = [ 1 1 0.3 ]:
[n, H] = householder([0 0 1]', [ 1 1 0.3 ]');
// "n" is the unit vector orthogonal to the reflecting plane

// Emitting object (feature from shell demo):
u = linspace(0,2*%pi,40);
v = linspace(0,2*%pi,20);
Xe = (cos(u).*u)'*(1+cos(v)/2)+10;
Ye = (u/2)'*sin(v);
Ze = (sin(u).*u)'*(1+cos(v)/2);

// Symetrical object:
Pe = [ Xe(:)' ; Ye(:)' ; Ze(:)'];
Pr = H*Pe;
Xr = matrix(Pr(1,:),40,-1);
Yr = matrix(Pr(2,:),40,-1);
Zr = matrix(Pr(3,:),40,-1);

// Reflecting plane containing O: n(1).x + n(2).y + n(3).z = 0
//   Sampling space:
x = linspace(min([Xe(:);Xr(:)]), max([Xe(:);Xr(:)]),20);
y = linspace(min([Ye(:);Yr(:)]), max([Ye(:);Yr(:)]),20);
[X, Y] = meshgrid(x,y);
//   Generating the mirror:
deff("z = mirror(x,y,n)","z = -n(1)/n(3)*x - n(2)/n(3)*y")
Zm = mirror(X,Y,n);

// Plotting:
clf()
isoview()
drawlater()
f = gcf();
f.color_map = [ 0.8 0.8 0.8 ; jet(100)];
surf(Xe,Ye,Ze)
surf(X,Y,Zm)
surf(Xr,Yr,Zr)
a = gca();
a.rotation_angles = [74 123];
a.children.color_flag = 0;
a.children.color_mode = 0;
a.children(1).foreground = color("red");
a.children(2).foreground = 1;
a.children(3).foreground = color("green");
drawnow()

See also

• proj — projection
• orthProj — Computes the orthogonal projection of a point to a polyline in the plane.
• scaling — affine transformation of a set of points
• qr — QR decomposition
• givens — Givens transformation

History