TUTORIAL

Graphing with 3DMath Explorer:

3DMath Explorer draws three dimensional (3D) graphs. But curve dimensions change with number of defined loop variables. So in this tutorial, in the first place we grouped curves with their dimensions as one dimensional (1D) curves, two dimensional (2D) curves and  three dimensional (3D) curves.

Now we take them one by one and tell how we draw these curves’ graphs with 3DMath Explorer. But please do remember 3DMath Explorer draws three dimensional (3D) graphs, and this don’t change with dimension of curves, 3DMath Explorer draws all kind of curves in the same 3D mathematical space.

In 3DMath Explorer, curves defines with 3 functions one for each 3 axis’s. For example to draw a simple function curve like y=2*x+4, blow definitions must be done.

fx= x

fy= 2*x+4

fz= 0              (=> in reality fz= 0*x=0 )

As you see x is the variable for three axis functions. So to draw this curve variable x must be defined as a loop variable with start, stop and step values.

-8<x<8   ,  Xstep =0.1

fx(x) = x

fy(x) = 2*x+4

fz(x) = 0

Now let’s enter this curves into curve explorer. As a result of this we get below graph.

6.2. Plotting Graphs of Two Dimensional (2D) Curves

Now let’s take a two dimensional (2D) curve and change same properties of it, and see how they would effect the graph (or what graphs we would get).

Sphere is a two dimensional curve. So it needs two loop variable that are valid for three axis to draw.

0<a<Pi     ,  a(step)=Pi/10

0<b<2*Pi  ,  b(step)=Pi/10

fx(a,b) = 5*sin(a)*cos(b)

fy(a,b) = 5*cos(a)

fz(a,b) = 5*sin(a)*sin(b)

When we chose the curve’s draw type as lines and its line withy as 8 on the curve explorer, we gets the below graph.

In the below, you see the same curve on a different background color after changing its line withy as 1 and its brush color as Lime.

Now let’s change start value of loop variable b from 0 to Pi/5 and rotate the graph a little by mouse.

Pi/5<b<2*Pi  ,  b(step)=Pi/10

Let’s remove fogging effect.

Now we activate fogging effect but remove perspective effect,

We again activate perspective effect and make visible yz-grid that covered with polygons

6.3. Plotting Graphs of Tree Dimensional (3D) Curves

Now let’s give a third dimension to a 2D sphere,

    0<a<Pi      ,  a(step)=Pi/10

Pi/5<b<2*Pi   ,  b(step)=Pi/10

fx(a,b) = 5*sin(a)*cos(b)

fy(a,b) = 5*cos(a)

fz(a,b) = 5*sin(a)*sin(b)

The constant 5 in the parameters of this curve is radius (r) of it. So we can change these expressions as below;

    0<a<Pi     ,  a(step)=Pi/10

Pi/5<b<2*Pi  ,  b(step)=Pi/10

    r = 5        (or 5<r<5   ,  r(step)=0)

fx(a,b,r) = r*sin(a)*cos(b)

fy(a,b,r) = r*cos(a)

fz(a,b,r) = r*sin(a)*sin(b)

Finally we have to define a new loop variable for third dimension. Now let’s define r as third dimension  variable. We made this variable r a loop variable with start value 4, end value 5 and step value 1, and we change step values of a and b with value 0.3.

    0<a<Pi       ,  a(step)=0.3

Pi/5<b<2*Pi   ,  b(step)=0.3

    4<r<5         ,  r(step)=1

fx(a,b,r) = r*sin(a)*cos(b)

fy(a,b,r) = r*cos(a)

fz(a,b,r) = r*sin(a)*sin(b)

6.4. Approaching to Curves / Zooming of Graphs

In reality, zooming of graphs mains looking them from a near position. Looking point is closely related with horizon point. When horizon point is selected as (0,0,z) and Z=-500, we are looking to the graph from coordinate at (0,0,z) and Z=+500. But in %100 zoom view, we wants 1 unit vertical distance to draw with 20 pixel. So this make us look to graph not from Z=500 but Z=25 (because 500/20 = 25) when %100 zoom view is selected. As a result when %200 zoom view  selected, our looking point came at coordinate (0,0,12.5), and when %400 zoom view selected, its goes to coordinate (0,0,6.25).

When perspective property of graph is on and the distance between our view point and graph origin is less than sphere’s radius, we would look and get view of graph from inside of the sphere towards to outside of it.

In the below we are looking at a graph from coordinate at (0, 0, 4) with %600 zoom. But this coordinate exists in a 3D sphere. This make us get a view of graph from inside of this 3D sphere. So we can see inside of this 3D sphere and  an other sphere that stays outside of this sphere.

Now let’s see striped view of the graph instead of the polygon view.

As you see, working on with 3DMath Explorer is such an easy. In the course of time, you get use to 3D math environment and will learn many more. But from now on, it  depends on you.