ARITH3(3)

💖 ARITH3(3) 💖 — operations on 3-d points and planes

okay bestie, welcome to the ARITH3(3) page from the Library Functions Manual ✨ this one is all about doing arithmetic on points and planes in affine or projective 3-space, and it is honestly lowkey iconic 🎀

🌸 NAME 🌸

add3, sub3, neg3, div3, mul3, eqpt3, closept3, dot3, cross3, len3, dist3, unit3, midpt3, lerp3, reflect3, nearseg3, pldist3, vdiv3, vrem3, pn2f3, ppp2f3, fff2p3, pdiv4, add4, sub4 — operations on 3-d points and planes 💅

🌸 SYNOPSIS 🌸

#include <draw.h>

#include <geometry.h>

Point3 add3(Point3 a, Point3 b)

Point3 sub3(Point3 a, Point3 b)

Point3 neg3(Point3 a)

Point3 div3(Point3 a, double b)

Point3 mul3(Point3 a, double b)

int eqpt3(Point3 p, Point3 q)

int closept3(Point3 p, Point3 q, double eps)

double dot3(Point3 p, Point3 q)

Point3 cross3(Point3 p, Point3 q)

double len3(Point3 p)

double dist3(Point3 p, Point3 q)

Point3 unit3(Point3 p)

Point3 midpt3(Point3 p, Point3 q)

Point3 lerp3(Point3 p, Point3 q, double alpha)

Point3 reflect3(Point3 p, Point3 p0, Point3 p1)

Point3 nearseg3(Point3 p0, Point3 p1, Point3 testp)

double pldist3(Point3 p, Point3 p0, Point3 p1)

double vdiv3(Point3 a, Point3 b)

Point3 vrem3(Point3 a, Point3 b)

Point3 pn2f3(Point3 p, Point3 n)

Point3 ppp2f3(Point3 p0, Point3 p1, Point3 p2)

Point3 fff2p3(Point3 f0, Point3 f1, Point3 f2)

Point3 pdiv4(Point3 a)

Point3 add4(Point3 a, Point3 b)

Point3 sub4(Point3 a, Point3 b)

🦋 DESCRIPTION 🦋

These routines do arithmetic on points and planes in affine or projective 3-space. Type Point3 is

typedef struct Point3 Point3;
struct Point3{
      double x, y, z, w;
};

Routines whose names end in 3 operate on vectors or ordinary points in affine 3-space, represented by their Euclidean (x,y,z) coordinates. They assume w=1 in their arguments, and set w=1 in their results. 🫶

✨ the ...3 routines ✨

• add3 — add the coordinates of two points.
• sub3 — subtract coordinates of two points.
• neg3 — negate the coordinates of a point.
• mul3 — multiply coordinates by a scalar.
• div3 — divide coordinates by a scalar.
• eqpt3 — test two points for exact equality.
• closept3 — is the distance between two points smaller than eps?
• dot3 — dot product.
• cross3 — cross product.
• len3 — distance to the origin.
• dist3 — distance between two points.
• unit3 — a unit vector parallel to p.
• midpt3 — the midpoint of line segment pq.
• lerp3 — linear interpolation between p and q.
• reflect3 — the reflection of point p in the segment joining p0 and p1.
• nearseg3 — the closest point to testp on segment p0 p1.
• pldist3 — the distance from p to segment p0 p1.
• vdiv3 — Vector divide — the length of the component of a parallel to b, in units of the length of b.
• vrem3 — Vector remainder — the component of a perpendicular to b. Ignoring roundoff, we have eqpt3(add3(mul3(b, vdiv3(a, b)), vrem3(a, b)), a).

🧚 plane + point conversions (the extra-cryptic names) 🧚

The following routines convert amongst various representations of points and planes. Planes are represented identically to points, by duality; a point p is on a plane q whenever p.x*q.x+p.y*q.y+p.z*q.z+p.w*q.w=0. Although when dealing with affine points we assume p.w=1, we can't make the same assumption for planes. the names of these routines are extra-cryptic — they contain an f (for face) to indicate a plane, p for a point and n for a normal vector. The number 2 abbreviates the word to. The number 3 reminds us, as before, that we're dealing with affine points. Thus pn2f3 takes a point and a normal vector and returns the corresponding plane. 💫

• pn2f3 — compute the plane passing through p with normal n.
• ppp2f3 — compute the plane passing through three points.
• fff2p3 — compute the intersection point of three planes.

💅 the ...4 routines 💅

The names of the following routines end in 4 because they operate on points in projective 4-space, represented by their homogeneous coordinates.

• pdiv4 — Perspective division. Divide p.w into p's coordinates, converting to affine coordinates. If p.w is zero, the result is the same as the argument.
• add4 — add the coordinates of two points.
• sub4 — subtract the coordinates of two points.

bestie, that p.w zero case in pdiv4 is the special little moment where the result is the same as the argument — no cap, that's literally what the page says 👑

🎀 SOURCE 🎀

/src/libgeometry

🌈 SEE ALSO 🌈

matrix(3)

and that's the ARITH3(3) footer moment: this page is from the Library Functions Manual, source /src/libgeometry, and there is no version/date stamped on it — just the manual name — so here's your cute undated sign-off, bestie 💖🫶✨