## Introduction

Normally, the Cartesian coordinate system is used in transformations and projections for graphics objects. In this case, you simply specify a point using `X`

, `Y`

, and `Z `

coordinates. In practice, other coordinate systems can also be applied, and are sometimes more convenient than the Cartesian coordinate system.

In this article, I will discuss the spherical coordinate system in 3D space and show you how to create the spherical graphics objects in this system.

## Background

In the spherical coordinate system, a point is specified by r, θ, and φ. Here r is the distance from the point to the origin, θ is the polar angle, and φ is the azimuthal angle in the `X`

-`Z`

plane from the `X `

axis. In this notation, I alternate the conventional `Y `

and `Z `

axes so that the computer screen is described by the `X`

-`Y `

plane. Figure 1 shows a point in this spherical coordinate system

Figure 1: Spherical coordinate system.

From this figure, we can obtain the following relationships:

The spherical coordinates (r, θ, φ) are related to the Cartesian coordinates by:

Sometimes it is more convenient to create sphere-like objects in terms of the spherical coordinate system. The following example application program will create two spheres. We need to add the `Matrix3 `

and `Point3 `

classes to the current project. And also add a new class, `DrawSphere`

, to the project.

Now we need to add a `Spherical `

method to the `Matrix3 `

class. The `Matrix3 `

class has been discussed in Chapter 5 of my new book "Practical C# Charts and Graphics".

public Point3 Spherical(float r, float theta, float phi)
{
Point3 pt = new Point3();
float snt = (float)Math.Sin(theta * Math.PI / 180);
float cnt = (float)Math.Cos(theta * Math.PI / 180);
float snp = (float)Math.Sin(phi * Math.PI / 180);
float cnp = (float)Math.Cos(phi * Math.PI / 180);
pt.X = r * snt * cnp;
pt.Y = r * cnt;
pt.Z = -r * snt * snp;
pt.W = 1;
return pt;
}

This method transforms a point in the spherical coordinate system to a point in the Cartesian coordinate system. We then Add a `DrawSphere `

class to the project. This class allows you to specify the radius and positions (the center location) of a sphere object. The `SphereCoordinates `

method in this class creates the points on a sphere surface by specifying their longitude and latitude. The `DrawIsometricView `

draws the sphere using the isometric projection.

## Using the code

This application can be tested using the following

`Form1 `

class:

using System;
using System.Drawing;
using System.Drawing.Drawing2D;
using System.Windows.Forms;
namespace Example5_6
{
public partial class Form1 : Form
{
public Form1()
{
InitializeComponent();
this.SetStyle(ControlStyles.ResizeRedraw, true);
panel1.Paint += new PaintEventHandler(panel1Paint);
}
private void panel1Paint(object sender, PaintEventArgs e)
{
Graphics g = e.Graphics;
g.SmoothingMode = SmoothingMode.AntiAlias;
float a = panel1.Height / 3;
DrawSphere ds = new DrawSphere(this, a, 0, 0, -a / 2);
ds.DrawIsometricView(g);
ds = new DrawSphere(this, 2 * a / 3, -a/2, -a/2, a / 2);
ds.DrawIsometricView(g);
}
}
}

Here we create two spheres with different radii and positions. By building and running this project, you should obtain the results shown in Figure 2.

Figure 2: Spheres created in a spherical coordinate system.

This project is from the examples (example5-6 in Chapter 5) of the new book "Practical C# Charts and Graphics", where you can find more advanced chart and graphics programming for real-world .NET applications. For more information, please visit the website at www.publishing.unicadinc.com

## About the Author

Dr. Jack Xu has a Ph.D in theoretical physics. He has over 15 years programming experience in Basic, Fortran, C, C++, Matlab, and C#, specializing in numerical computation methods, algorithms, physical modeling, computer-aided design (CAD) development, graphics user interface, and 3D graphics. Currently, he is responsible for developing commercial CAD tools based on Microsoft .NET framework.

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