Three Dimensional Drawing ("3D drawing)

Drawing a simple box in perspective provides good practice in drawing geometric figures. Also, the confidence studentswill gain from these projects, will help them to draw other geometric solids. This will be helpful when studying volume andsurface area. It will greatly enhance the students' ability to understand, draw, and solve problems in the volume chapter of

Geometry. Students will find it very helpful to them, if they have learned to draw geometric solids such as prism, pyramid,and cone.

There are a number of different ways to draw "three dimensional drawings". The easiest way is called an "ObliqueDrawing". In an oblique drawing, the front of the object is drawn "head on", as if it were facing the viewer; then the sidesof the object recede at a 45 degree angle, as shown in the oblique drawing below:

The second type of 3D drawing is called "Isometric Drawing". In an isometric, there are no horizontal lines. As you see in

the Isometric Drawing below, vertical lines are vertical, but the angled lines are both at a 30 degree angle, one going tothe left and the other to the right.

The main advantage of drawing in oblique is that it is the easiest way to draw something in three dimensions. The frontface is "flat", so you simply draw a square. Then the three vertical edges of the cube are drawn as vertical lines. The

receding lines (the ones going back in space) are drawn at 45 degree angles, parallel to each other as shown. In thisdrawing, the receding lines are drawn a bit shorter than the the vertical or horizontal edges, so that it will look "right" tothe viewer. In "real life" objects look smaller whe they are further from the viewer.

Another way to draw, is in Perspective. Not only is this an interesting and "fun" project, but it is a useful skill to learn.Many students say that they "can't draw", but this simple project may change their attitude about that, as it is fun andeasy. It might be best to start them off with drawing a box in perspective as shown below, following the instructions, and

then draw some other geometric solids.

The box below is drawing in "One Point Perspective". The front is drawn facing forward, and is a square. The top andright side recede to one vanishing point, in the distance. This vanishing point may be selected at random.

The drawing below shows a "Two Point Perspective of the box. Notice that there are no horizontal lines. The left face ofthe box recedes to a vanishing point on the left, and the right face of the box recedes to a vanishing point on the right.Notice that the two vanishing points are on the same horizontal line (although that line is not drawn). This is essential, in a2-point perspective.

A project that students will enjoy is drawing a word in perspective. The example below is a one-point perspective:

To construct a one point perspective such as the one above, follow these steps:

1) Draw a horizontal line at the bottom; this is called the Ground Line. Pick a point above the Ground Line to be theVanishing point, as shown above. In a one-point perspective, there are two types of lines: horizontal lines, and lines thatgo to the vanishing point.

2) Now let's use the letter T as an example. Notice that all the vertical lines in the word are vertical in the perspectivedrawing, and horizontal lines are horizontal in the ) drawing. But the lines that give the drawing its depth are all drawn tothe vanishing point as shown. This is what makes this a perspective drawing. the depth (or "thickness) of each lettershould be the same, in a one-point perspective.

With these simple steps students can draw the letters of their name, or can be assigned a mathematical term to drawthemselves. If students would like to know more about perspective, they can go to the following web page:http://mathforum.org/workshops/sum98/participants/sanders/Persp.html

Perspective drawing is just one of a number of ways to draw objects that look 3-dimensional. Two other types of "3D"

drawings are called "oblique" and "isometric". The drawing below is an oblique drawing: all lines are either horizontal,vertical, or receding at a 45 degree angle. The first image below is an "Oblique grid". Students can place a sheet oftracing paper over either one of the two grids, and use the lines of the grid to help them to create a 3-dimensionalOblique drawing, as in the second drawing below:

A mathematical 3-D maze is another interesting project for students to create, as shown below. Is this an oblique,

isometric, or a perspective drawing? You will find the answer to this question near the end of this web page.

The mathematical 3D maze was drawn in isometric. How can you tell? The lines appear to be at 30 degree angles to the

left and right, and since they are parallel it can't be a Perspective; there are two sets of lines, one going at an angle to theleft and the other at an angle to the right, so it is not an Oblique. Therefore it must be an Isometric drawing!

Students usually find the topic of perspective drawing very interesting, and really enjoy creating their own one- and two-point perspectives. The more interesting a student finds mathematics, the more likely the student is to appreciate math.

http://mathforum.org/workshops/sum98/participants/sanders/Persp.html

And every teacher knows that an actively involved student is a student who will learn (and retain) more mathematics!

Geometry enlightens the intellect and sets one's mind right. All of its proofs are very clear andorderly. It is hardly possible for errors to enter into geometrical reasoning, because it is well arrangedand orderly. Thus, the mind that constantly applies itself to geometry is not likely to fall into error. Inthis convenient way, the person who knows geometry acquires intelligence.

Ibn Khaldun (1332-1406

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http://mathforum.org/sanders/creativegeometry/incontents.htm