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hyperfun:tut_trans

## Let's try to shift, scale, rotate objects

### Let's try to shift objects

Array x in my_model(x[3], a[1]) includes coordinates in three dimensions. The corresponding relationship is,

```x[1] -> x
x[2] -> y
x[3] -> z```

The following is an equation of a solid sphere which has its center as the origin and a radius of 5: The model in HyperFun:

```my_model(x[3], a[1])
{
sphere = 5.0^2 - (x[1]^2 + x[2]^2 + x[3]^2);

my_model = sphere;
}```

What is the difference between these two equations: and ? We will experiment with two equations in HyperFun.

```my_model(x[3], a[1])
{
sphere = 5.0^2 - ((x[1]-5.0)^2 + x[2]^2 + x[3]^2);

my_model = sphere;
}```

Then what is the difference between these two equations: and ? We will experiment with two equations in HyperFun.

```my_model(x[3], a[1])
{
sphere = 5.0^2 - ((x[1]+5.0)^2 + x[2]^2 + x[3]^2);

my_model = sphere;
}```

The result of experiments with HyperFun shows us that replacing x with x-5 is equal to moving only 5 in the positive direction of the x-axis, and replacing x with x+5 is equal to moving only 5 in the negative direction of the x-axis.

Question:

Given the equation , how is this equation moved only 5 in the positive direction of the x-axis?

Let's try to shift the sphere in the direction of the y-axis, the z-axis. In HyperFun, there is only one operation for shifting in 3D space: hfShift3D

### Let's try to scale objects

First let's make a sphere, which has its center as the origin and a radius of 5.

```my_model(x[3], a[1])
{
sphere = 5.0^2 - (x[1]^2 + x[2]^2 + x[3]^2);

my_model = sphere;
}```

What is the dirference between these two equations: and ? We will experiment in HyperFun, replacing x[1] with x[1]/2.

```my_model(x[3], a[1])
{
sphere = 5.0^2 - ((x[1]/2)^2 + x[2]^2 + x[3]^2);

my_model = sphere;
}```

What is the dirference between these two equations: and ? We will experiment in HyperFun, replacing x[1] with x[1]*2.

```my_model(x[3], a[1])
{
sphere = 5.0^2 - ((x[1]*2)^2 + x[2]^2 + x[3]^2);

my_model = sphere;
}```

As the result of experimenting with HyperFun, a sphere is scaled twice with the x-axis, replacing x with x/2, a sphere is scaled 1/2 times with the x-axis, replacing x with x*2. Let's try to scale the sphere with y-axis, z-axis in the same way.

In HyperFun, there is only one operation for scaling: hfScale3D

### Let's try to rotate objects

Rotating at s radian in three dimension around the z-axis is described in HyperFun as follows, (s radian is pi*s/180 degree.)

```x' = x cos(s) + y sin(s)
y' = y cos(s) - x sin(s)
z' = z```

We will experiment with the upper expressions in HyperFun.

```my_model(x[3], a[1])
{
array xt[3];
pi = 3.14159;

sphere1 = 3.0^2 - ((x[1] - 5)^2 + x[2]^2 + x[3]^2);

xt[3] = x[3];

sphere2 = 3.0^2 - ((xt[1] - 5)^2 + xt[2]^2 + xt[3]^2);

my_model = sphere1 | sphere2;
}```

The sphere1 is a sphere which has the center of (5, 0, 0) and a radius of 3. The sphere2 is the sphere1 which is rotated at a 90 angle around the z-axis. As the result of using the upper expressions, we can make a sphere rotate at a 90 angle around the z-axis. The following expressions are ones of rotation around y-axis and z-axis.

Let's try to use them.

Expressions of rotation around y-axis.

```x' = x cos(s) - z sin(s)
y' = y
z' = x cos(s) + z sin(s)```

Expressions of rotation around x-axis.

```x' = x
y' = y cos(s) + z sin(s)
z' = z cos(s) - y sin(s)```

In HyperFun, there are three operations for rotation: hfRotate3DZ (rotation around z-axis), hfRotate3DY (rotation around y-axis), hfRotate3DX (rotation around x-axis).