# Introduction

*geometry*above an art work. We have been describing the

*Euclidean*geometry as the most common geometry used in art.

In lesson #2 we have bypass the *Euclidean* geometry and introduce a new one: the *spherical* geometry which gives more FOV power to the artist. So the next time you’ll get to your preferred *plein air* location or *urban*location, I want you to ask your self *how much FOV I want to sketch?* If you only want to shot against a flower vase it is ok to choose *Euclidean*geometry, but, I you want to draw what is happening in your common cafe, things came difficult to achieve in *Euclidean* geometry. You have to shift into *spherical* geometry.

What is important to retain, as I explain in lesson 1 is that below any drawing, illustration, sketch you draw, lays a geometry. Do your remember when you build a perspective grid? (I guess you’ve done that). Well, perspective grid is nothing but a geometry. What I am trying to say, is: the election of a particular geometry comes to join the election of color harmony, shapes, lights and shades as a **new aesthetic criterion** to build an artwork. I will repeat this as many times is necessary: The way you manage your *space* in your drawing pad is intimately related with the *geometry* chosen.

Today we are going to concentrate our energies in the *equirectangular* grid which is the flat support which bring us all the power of the *spherical*geometry.

The purpose of this lesson is to understand how the equirectangular works and how we can get the maximum profit in our next drawings.

Let’s take the surfboard and dive into the Spherical Ocean with the amazing equirectangular waves!

### Vanishing Points

What happens when you look in front of you? Have you ever asked yourself this question? I want you to make the following visual exercise, does not matter where you are, indoors out outdoors:

Look in front of you and try to find out where the horizontal lines vanishes. You got it? These lines vanishes in a point located in front of your nose and intersect with the horizon line.

We can then build a *central* perspective grid like this one:

And it is very easy to draw objects with a grid like this, we’ve done that at school in art classes.

You just have to try to follow the line that escapes towards the central point and outline linear segments within this chosen line.

Now, want happens when we rotate our head to the left or the right?

We see again how the objects are organized in the same point central grid and how the horizontal lines vanishes into a central point on the horizon line. Wherever we look, the same thing always happens: a central point where everything converges. Please do the following exercise: Look to your right, then to your left, then turn the head and look to the sky, and finally look at your feet. And takes five seconds to try to detect where that central point is located.

It’s as if we had a wire mesh welded to the head, funny isn’t?

### The geodesic connection

Let’s go back to our old master painter friend. In front of him stands a stoned wall, like shown in the next illustration. When he look to the front, the wall stands parallel to him and the central point is located in front of him at the horizon line. When he look to the left or his right, the wall vanishes to a central point every time as we have describe above.

Our friend is wondering how to draw the three vanishing point in the same sheet of paper.

Some artist has a great art work, and sometimes great art work makes their ego boost. And, I don’t know why, when the ego boost, ears shut down, maybe Freud has an explanation. So, my old Master Friend painter, has completely forgotten my last class. So, let’s recall it:

We observe two things: the vanishing points are located on the surface of the sphere. The three vanishing points are connected by a very special curve, ladies and gentlemen, this is curve is called a **Geodesic**. The geodesic goes through the surface of the sphere connecting the three points mentioned. And the path which connect this three points is also a minimal path.

Geodesic are present every day in our life and have fascinated physicists and mathematicians for centuries. When you take a plane or a boat the path you travel is a geodesic. When you open the window in the morning and let the sun come into your bedroom, photons travel in geodesics lines. This adorable lady deserves all our attention and be drawn.

### The E.G.G

Please pay attention now, things are finally coming out. The chicken has pound an **EGG** ( E.G.G stands for *Equirectangular Geodesic Grid*).

We are going to describe in detail our new friend: the **Egg**

We have 6 vanishing points in our Egg.

- Front, just in the middle
- Right, at 90º from the front
- Left at -90º from the front
- Back vanishing point which is half on the left side and half on the right side
- finally, up and down. See those to thick lines next to the border of the Egg? Those are up and down vanishing points. Don’t worry about up and down vanishing points, we will dedicate the next lesson to this special guys.

Let’s see how can we sketch simple objects in the Egg file.

What you see in front of you will be located in the front area. What you have on the right, in the area on the right, what you have on the left in the area on the left. Once you finish to sketch a face, turn your head 90º so that you have a nice central point perspective. Now, locate the central point perspective in the egg and try to find the geodesics where objects are supported, in the same way we did with our linear central point perspective:

If you want to have a 360º visual experience, I have uploaded this egg into a VR support, here. Have you notice how the geodesic lines suddenly became linear straight lines? This is the magic of mathematics, playing with the space as an illusionist. Bored a Sunday afternoon? Open a math book!

### Conclusions

In this lesson, we started to introduce the *equirectangular geodesic grid*(egg) and its powerful properties. Through the egg we can connect different vanishing points. This connection is establish through a geodesic curve. The egg can give us the possibility to extend the FOV up to 360º horizontal and 180º verical. However, this is not mandatory. We can use the egg with a FOV of 180º or less, as it is shown in this drawing. In the next lesson, we will study the projection of objects close to up and down poles. Since equirectangular projection has a different behavior when we get close to north and south poles, I rather prefer to dedicated a particular lesson to it and keep in this paper essential concepts.

Thank you for reading and keep tuned to this course!