Points vs vectors

Both are written as a list of numbers, such as (6, 3), so it is easy to treat them as the same thing. They are not. A point is a position: the start of a wall, the centre of a column, the corner of a slab. A vector is a displacement: “6 m east and 3 m north”. It has a direction and a length, but no position. You can draw the same vector starting anywhere.

Drag A and B, then move both together with the buttons. The coordinates of the points change; the vector between them does not.

The four rules

There is one useful exception to the last rule. A weighted average of points is fine as long as the weights add up to 1. The midpoint is ½A + ½B, which is really A + ½(B − A): a point plus a vector. The same idea gives any point along a wall: A + t(B − A), where t = 0 is A and t = 1 is B.

Points in a BIM model, and vectors

The difference shows up as soon as you move something. Moving a building 10 m east changes every point in it, but a wall’s direction, a face’s normal and the direction to the sun stay exactly the same. A rotation, on the other hand, changes both. The Matrices module uses this: points and vectors are transformed by different rules, and mixing them up shifts normals off their faces.

Let the compiler keep them apart

Many APIs use one type for both (the Revit API’s XYZ, System.Numerics.Vector3), so nothing stops you from adding two points. If you write your own types, you can make the four rules the only operations that compile:

1var a = new Pt(2, 1);2var b = new Pt(8, 4);3 4Vec v = b - a;                        // point - point = vector5Console.WriteLine($"v = {v}");6 7var east = new Vec(10, 0);8Pt a2 = a + east, b2 = b + east;      // point + vector = point9Console.WriteLine($"Moved: {a2} to {b2}");10Console.WriteLine($"v after the move = {b2 - a2}");11 12// Pt nonsense = a + b;               // does not compile: no Pt + Pt operator13 14readonly record struct Pt(double X, double Y)15{16    public static Vec operator -(Pt a, Pt b) => new(a.X - b.X, a.Y - b.Y);17    public static Pt operator +(Pt p, Vec v) => new(p.X + v.X, p.Y + v.Y);18    public override string ToString() => $"({X}, {Y})";19}20 21readonly record struct Vec(double X, double Y)22{23    public override string ToString() => $"<{X}, {Y}>";24}

v = <6, 3>
Moved: (12, 1) to (18, 4)
v after the move = <6, 3>

1var start = new Pt(1, 2);2var end = new Pt(9, 6);3Vec span = end - start;4 5for (int i = 1; i <= 3; i++)6    Console.WriteLine($"Hanger {i}: {start + span * (i / 4.0)}");7 8readonly record struct Pt(double X, double Y)9{10    public static Vec operator -(Pt a, Pt b) => new(a.X - b.X, a.Y - b.Y);11    public static Pt operator +(Pt p, Vec v) => new(p.X + v.X, p.Y + v.Y);12    public override string ToString() => $"({X}, {Y})";13}14 15readonly record struct Vec(double X, double Y)16{17    public static Vec operator *(Vec v, double k) => new(v.X * k, v.Y * k);18}

Hanger 1: (3, 3)
Hanger 2: (5, 4)
Hanger 3: (7, 5)