03 Transforms.dib

#!meta

{"kernelInfo":{"defaultKernelName":"csharp","items":[{"aliases":[],"name":"csharp"}]}}

#!markdown

### Setup

#!csharp

#r "nuget: Hypar.Elements, *-*"
List<Line> FinalHypar() {
    var line = new Line((-1,0,0), (1,0,0));
    var line2 = new Line((0, -1, 0), (0, 1, 0));
    var hyparLines = new List<Line>();
    for(int i=-10;i<10;i++) {
        var rotationAboutYAxis = new Transform().Rotated(Vector3.YAxis, i*2);
        var translationAlongYAxis = new Transform().Moved(0,i*0.1,0);
        var transform = rotationAboutYAxis.Concatenated(translationAlongYAxis);
        var transformedLine = line.TransformedLine(transform);
        hyparLines.Add(transformedLine);

        var rotationAboutXAxis = new Transform().Rotated(Vector3.XAxis, -i*2);
        var translationAlongXAxis = new Transform().Moved(i*0.1,0,0);
        var transform2 = rotationAboutXAxis.Concatenated(translationAlongXAxis);
        var transformedLine2 = line2.TransformedLine(transform2);
        hyparLines.Add(transformedLine2);
    }
    return hyparLines;
}

#!markdown

# Transforms

Moving, rotating, and scaling geometry is achieved through the use of `Transform`s. These are a common concept in computer graphics, but they take a little getting used to. You can think of a transform as an object which contains a set of "instructions" for how to modify the scale, rotation, and position of any element it is applied to. It's also sometimes useful to think of a transform as an oriented coordinate frame, similar to a `Plane` in Grasshopper.

We're using a special trick here to render them as a set of axes. The transform constructed with `new Transform()` is located at (0,0,0), with its X, Y, and Z axes oriented along the world X, Y, and Z axes.

#!csharp

var transform = new Transform();
return transform;

#!markdown

There are a number of `Transform` constructors available:

`new Transform()` - 
`new Transform(3, 4, 5)` - 
`new Transform(new Vector3(2,1,7), new Vector3(1,1,1))` - Create a transform that originates at (2,1,7), with its normal (Z-axis) pointing along (1,1,1)
`new Transform(new Vector3(1,2,4), new Vector3(0,1,0), new Vector3(1,0,1))` 

#!csharp

// Create the identity transform (no change / world origin)
var origin = new Transform();
// Create a world-aligned transform at the position (2, 2, 2) (applying this transform can also be thought of as "moving" an object from 0,0,0 to 3,4,5)
var positionedTransform = new Transform(2, 2, 2);
// Create a transform that originates at (2,1,7), with its normal (Z-axis) pointing along (1,1,1)
var orientedTransform = new Transform(new Vector3(5, 1, 3), new Vector3(1, 1, 1));
// Create a transform that originates at (7,2,4), with its X-axis pointing along (0,1,0), and its Z-axis pointing along (-1,0,1)
var alignedTransform = new Transform(new Vector3(7, 2, 4), new Vector3(0, 1, 0), new Vector3(-1, 0, 1));
return new [] { origin, positionedTransform, orientedTransform, alignedTransform };

#!markdown

Often it's easier to apply sequential transformations to an element, rather than trying to construct a single transform that does everything you want. 
`Elements` supports two versions of all transform methods: one that modifies the current transform (like `Move`, `Rotate`, and `Scale`,) and one that makes a modified copy of it (like `Moved`, `Rotated`, and `Scaled`).

For example, if you want to move an object 5 units in the X direction, then rotate it 45 degrees around the Z axis, you can construct a transform like this:

#!csharp

var origin = new Transform();
var transform = new Transform().Moved((5,0,0)).Rotated(Vector3.ZAxis, 45);
return new [] {origin, transform};

#!markdown

OK, that's enough floating axes in space. Let's see how to use transforms.

We can modify profiles, polygons, lines, and other curves using their `Transformed` methods.

#!csharp

var star = Polygon.Star(5,2,7);
var starMoved = star.TransformedPolygon(new Transform().Moved(5,1,3));
return new [] {star, starMoved};

#!csharp

var line = new Line((-1,0,0), (1,0,0));
var hyparLines = new List<Line>();
for(int i=-10;i<10;i++) {
    var rotationAboutYAxis = new Transform().Rotated(Vector3.YAxis, i*2);
    var translationAlongYAxis = new Transform().Moved(0,i*0.1,0);
    // use "Concatenated" to combine the two transforms
    var transform = rotationAboutYAxis.Concatenated(translationAlongYAxis);
    var transformedLine = line.TransformedLine(transform);
    hyparLines.Add(transformedLine);
}
return hyparLines;

#!markdown

# Exercise

See if you can construct the "other axis" of the Hyperbolic Paraboloid seen above, and display both: 

#!csharp

// Your end result should look like this:
return FinalHypar();

#!csharp

// Your code here: