Ao.Geometry

The Ao.Geometry namespace contains classes for geometric calculations in 2D and 3D.

Linear Equations

Linear equations play an important role in Euclidean geometry and computer graphics. They represent straight lines in 2D and planes in 3D. Several forms of linear equations exist.

The General structs represent the general form.

\[0 = a \cdot x + b \cdot y + c\] \[0 = a \cdot x + b \cdot y + c \cdot z + d\]

var G = new General2
{
    A = 1,
    B = 1,
    C = -2
};

Console.WriteLine(G.InterceptX);
Console.WriteLine(G.InterceptY);

2
2

The Hesse structs represent the Hesse normal form.

\[d = \vec{p}\cdot\vec{n}\]

var H = new Hesse2
{
    D = 4,
    Normal = new Vector2(2, 2)
};

Console.WriteLine(H.InterceptX);
Console.WriteLine(H.InterceptY);

2
2

The Line2 and Plane3 structs represent the respective parametric form.

\[\vec{x}(t) = \vec{b} + t \cdot \vec{d}\] \[\vec{x}(u, v) = \vec{b} + u \cdot \vec{d_1} + v \cdot \vec{d_2}\]

var L = new Line2
{
    Base = new Point2(-2, 4),
    Direction = new Vector2(3, -3)
};

Console.WriteLine(L.InterceptX);
Console.WriteLine(L.InterceptY);

2
2

The forms can be derived from each other as well as from other forms, such as the slope-intercept form in 2D or the intercept form.

var G = General2.FromIntercepts(2, 2);

Console.WriteLine(G.A);
Console.WriteLine(G.B);
Console.WriteLine(G.C);

2
2
-4

Additionally, the calculation of intersections and distances is supported.

var d = G.Distance(Point2.Origin);

Console.WriteLine(d);

1.41421356237309

Affine Transformations

The Affinity classes represent affine transformations. An affine transformation consists of a transformation matrix and a translation vector. Several constants and static methods provide common transformations, that are often used in computer graphics.

The following example demonstrates how to rotate a point about 180° around the z-axis.

var T = Affinity3.RotateZ(Math.PI);

var P = new Point3(1, 2, 3);

var Q = T * P;

Console.WriteLine("P = ({0,2}, {1,2}, {2,2})", P.X, P.Y, P.Z);
Console.WriteLine("Q = ({0,2}, {1,2}, {2,2})", Q.X, Q.Y, Q.Z);

P = ( 1,  2,  3)
Q = (-1, -2,  3)

Also, affine transformations can be combined …

var T1 = Affinity3.RollPitchYaw(Math.PI, Math.PI / 2, 0);
var T2 = Affinity3.Translate(2, -5, 8);

var T = T2 * T1;

… and inverted.

var TI = T.Inverted;