Introduction to path surfaces

Introduction to path surfaces#

Borromean rings, A, B, C#

Some text explaining the importance of building the mesh in x-direction.

An example in C#, for A

var r = Math.Sqrt(3) / 3;
var x = Math.Cos(t);
var y = Math.Sin(t) + r;
var z = Math.Cos(3 * t) / 3;

An example in C#, for B

var r = Math.Sqrt(3) / 3;
var x = Math.Cos(t) + 0.5;
var y = Math.Sin(t) - r / 2;
var z = Math.Cos(3 * t) / 3;

An example in C#, for C

var r = Math.Sqrt(3) / 3;
var x = Math.Cos(t) - 0.5;
var y = Math.Sin(t) - r / 2;
var z = Math.Cos(3 * t) / 3;

Ellipses, A, B, C#

Some text explaining the importance of building the mesh in x-direction.

An example in C#, for A, B, C

var x = 2 * Math.Cos(t);
var y = Math.Sin(t);
var z = 0.0;

For A, the final rotations are: X=0, Y=0, Z=0.

For B, the final rotations are: X=90, Y=0, Z=90.

For C, the final rotations are: X=0, Y=90, Z=90.

Trefoil 2#

Some text explaining the importance of building the mesh in x-direction.

An example in C#, for A, B, C

var D = 2.0; //  D = 1.0;  D = 2.0;
var x = D * Math.Sin(t) + 2 * Math.Sin(2 * t);
var y = D * Math.Cos(t) - 2 * Math.Cos(2 * t);
var z = -D * Math.Sin(3 * t);

Formatting options for path surfaces, Trefoil 5#

These are the rough versions

An example in C#, for A, B, C

var D = 1.0; //  D = 1.0;  D = 2.0;
var x = D * Math.Sin(t) + 2 * Math.Sin(2 * t);
var y = D * Math.Cos(t) - 2 * Math.Cos(2 * t);
var z = -D * Math.Sin(3 * t);

TestALines_Trefoil05_a \(\quad\) TestALines_Trefoil05W_a \(\quad\) TestALines_Trefoil05WO_a

Left figure: Trefoil, rough version (see also Wikipedia [1462], Gray et al. [368], Krivoshapko and Ivanov [421]). Camera angles are \(\theta=135^\circ\) and \(\phi = -12^\circ\).

Middle figure: Trefoil, rough version (see also Wikipedia [1444], Gray et al. [368], Krivoshapko and Ivanov [421]). Camera angles are \(\theta=135^\circ\) and \(\phi = -12^\circ\).

Right figure: Trefoil, rough version (see also Wikipedia [1444], Gray et al. [368], Krivoshapko and Ivanov [421]). Camera angles are \(\theta=135^\circ\) and \(\phi = -12^\circ\).

These are the smooth versions

TestALines_Trefoil05S_a \(\quad\) TestALines_Trefoil05WS_a \(\quad\) TestALines_Trefoil05WSO_a

Left figure: Trefoil, smooth version (see also Wikipedia [1462], Gray et al. [368], Krivoshapko and Ivanov [421]). Camera angles are \(\theta=135^\circ\) and \(\phi = -12^\circ\).

Middle figure: Trefoil, smooth version (see also Wikipedia [1444], Gray et al. [368], Krivoshapko and Ivanov [421]). Camera angles are \(\theta=135^\circ\) and \(\phi = -12^\circ\).

Right figure: Trefoil, smooth version (see also Wikipedia [1444], Gray et al. [368], Krivoshapko and Ivanov [421]). Camera angles are \(\theta=135^\circ\) and \(\phi = -12^\circ\).

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