Miscellaneous Geometric Functions

vplus vector-list [function]

returns a newly created float-vector that is the sum of all the elements of vector-list. The difference from v+ is that vplus computes the sum of more than two arguments and no result vector can be specified.

vector-mean vector-list [function]

returns the mean vector of vector-list.

triangle a b c &optional (normal #f(0 0 1)) [function]

a, b, c are float-vectors representing 2 or 3 dimensional points. normal is the normal vector of the plane on which a,b, and c lie. Triangle returns 2*area of a triangle formed by a,b,c. Triangle is positive if a,b, and c turn clockwise when you are looking in the same direction as normal. In other words, if triangle is positive, c locates at the left hand side of line a-b, and b lies at the right side of ac.

triangle-normal a b c [function]

finds a normal vector which is vertical to the triangle defined by three points a,b,and c.

vector-angle v1 v2 &optional (normal (v* v1 v2)) [function]

Computes an angle between two vectors, denoted by $atan(normal \cdot (v1 \times v2), v1 \cdot v2)$. v1,v2 and normal must be normalized vectors. When normal is not given, a normalized vector commonly perpendicular to v1 and v2 is used, in which case the result is always a positive angle in the range between $0$ and $\pi$. In order to obtain a signed angle, normal must be specified explicitly.

face-normal-vector vertices [function]

Computes surface normal vector from a list of float-vectors which lie on the same plane.

farthest p points [function]

finds the farthest point from p in the list of 3D float-vectors, points.

farthest-pair points [function]

finds the farthest point pair in the list of 3D float-vectors, points.

maxindex 3D-floatvec [function]

Finds the index of the absolute maximum value of three elements.

random-vector &optional (range 1.0) [function]

Generates a random vector which is distributed homogeneously in 3D Cartesian space.

random-normalized-vector &optional (range 1.0) [function]

returns a normalized-3D random vector.

random-vectors count range [function]

returns a list of random vectors.

line-intersection p1 p2 p3 p4 [function]

p1, p2, p3 and p4 are all float-vectors of more than two dimensions. p1-p2 and p3-p4 define two lines on a plane. line-intersection returns a list of two parameters of the intersection point for these two lines. When used in three dimension, p1, p2, p3 and p4 must be coplanar.

collinear-p p1 p2 p3 &optional tolerance [function]

p1, p2, p3 are all three-dimensional float-vectors representing three point locations. Collinear-p returns the parameter for p2 on the line p1-p3 if $norm((p2-p1) \times (p3-p1))$ is smaller than *coplanar-threshold*, otherwise NIL.

find-coplanar-vertices p1 p2 p3 vlist [function]

p1, p2, p3 are all three-dimensional float-vectors representing a plane. Find-coplanar-vertices looks for coplanar points in vlist that lie on the plane.

find-connecting-edge vertex edgelist [function]

finds an edge in edgelist that connects to vertex.

make-vertex-edge-htab bodfacs [function]

bodfacs is a body or a list of faces. make-vertex-edge-htab makes a hash-table which allows retrieving of edges connected to a vertex.

left-points points p1 p2 normal [function]

Assume points, p1, and p2 lie on the plane whose normal vector is normal. Left-points searches in points and collects ones lying in the left hand side of the line passing on p1, p2.

right-points points p1 p2 normal [function]

Assume points, p1, and p2 lie on the plane whose normal vector is normal. Right-points searches in points and collects ones lying in the right hand side of the line determined by p1, p2.

left-most-point points p1 p2 normal [function]

Assume points, p1, and p2 lie on a plane whose normal vector is normal. left-points searches in points which lie in the left-hand side of the line determined by p1, p2 and returns the farthest one.

right-most-point points p1 p2 normal [function]

Assume points, p1, and p2 lie on a plane whose normal vector is normal. right-most-point searches in points which lie in the right-hand side of the line determined by p1, p2 and returns the farthest one.

eps= num1 num2 [(tolerance *epsilon*)] [function]

compares two float numbers num1 and num2 for equality with the tolerance of *epsilon*.

eps$<$ num1 num2 [(tolerance *epsilon*)] [function]

returns T if $num1$ is apparently less than $num2$, i.e. $num1 < num2-tolerance$.

eps$<=$ num1 num2 [(tolerance *epsilon*)] [function]

returns T if $num1$ is possibly less than or equal to $num2$, i.e. $num1 < num2+tolerance$.

eps$>$ num1 num2 [(tolerance *epsilon*)] [function]

returns T if $num1$ is apparently greater than $num2$, i.e. $num1 < num2-tolerance$.

eps$>=$ num1 num2 [(tolerance *epsilon*)] [function]

returns T if $num1$ is possibly greater than or equal to $num2$, i.e. $num1 < num2+tolerance$.

bounding-box [class]

  :super   object 

  :slots 		 (minpoint maxpoint) 

defines a minimal rectangular-parallel-piped which is bounded by the planes parallel to xy-, yz- and zx-planes. Bounding-box can be used in any dimension according to the dimension of vectors given at the initialization. Bounding-box had been defined by the name of surrounding-box.

:box [method]

returns this bounding-box object itself.

:volume [method]

returns the volume of this bounding box.

:grow rate [method]

increases or decreases the size of this box by the rate. When rate is 0.01, the box is enlarged by 1%.

:inner point [method]

returns T if point lies in this box, otherwise nil.

:intersection box2 &optional tolerance [method]

returns the intersectional bounding box of this box and box2. If tolerance is given, the box is enlarged by it. If there is no intersection, NIL is returned.

:union box2 [method]

returns the union of bounding box of this box and box2.

:intersectionp box2 [method]

returns T if this box has the intersection with the box2, NIL otherwise. This method is faster than :intersection because no new instance of bounding-box is created.

:extreme-point direction [method]

returns one of the eight corner points yielding the largest dot-product with direction.

:corners [method]

returns the list of all vertices of this box. If this box defines 2D bounding-box, then 4 points are returned, 3D, 8, and so on.

:below box2 &optional (direction #(0 0 1) [method]

returns T if this box is below box2 in direction. This is used to check whether two box intersects when this box is moved toward direction.

:body [method]

returns a body object that represents a cube bounded by this box.

:init vlist &optional tolerance [method]

sets minpoint and maxpoint slots looking in vlist. If tolerance (float) is specified, the box is grown by the amount.

make-bounding-box points [tolerance] [function]

finds the minimum and maximum coordinates in the list of points, and make an instance of bounding-box.

bounding-box-union boxes [tolerance *contact-threshold*] [function]

makes an instance of the surrounding-box representing the union of boxes. The resulted box is expanded by the tolerance.

bounding-box-intersection boxes [tolerance *contact-threshold*] [function]

makes an instance of the surrounding-box representing the intersection of boxes. The resulted box is expanded by the tolerance.