原臺灣大學土木工程學系教授 (Professor) 土木工程學系 (Department of Civil Engineering)

Hello, everyone, today's topic is "Circles, Ellipses and Arcs”.

These are three key notes to this lesson. First is how to draw circles.

There are some basic methods, three common methods are: Square method, Isometric-assisted method and Pivot method.

I will introduce and demonstrate these methods to you later.

As for ellipses, the concept is simple, it is a transformation of a circle,

so all we need to do is adjust the way we draw circles a bit and we can draw ellipses.

As for arcs, it is, in fact, a section of a circle, so it can be drawn with the same method. I will give a more detailed introduction later.

Well, we can see here, this is a circle drawn by computer.

It is perfect. We know that circles have many mathematical and geometrical properties.

For example, we all know that a circle has a center, and the distance between the center and any point on the circle, that is, the radius, is exactly the same.

The distance between the center and any point on the circle is exactly the same.

Therefore, when we are drawing a circle, we can use such a geometrical property.

Now, I would like to first introduce you the Square method. To draw circles, you can draw a square and fit a circle into it.

First, you draw the diagonals in a square, then, you will be able to find the center for a circle, right?

When you want to draw a circle, but you don't know where to start,

the simplest method is to draw a square, then, the diagonals, and you will find the center of the circle.

When you are drawing the circle freehand, and you don't have a compass,

a very simple way to draw the circle is to start from the juncture of the square and the circle.

And gradually connect the junctures and construct a circle. This is the Square method.

Well, this square, we take one side, which is equal to the diameter of the circle, this is a geometric property that we can use.

We draw very faint lines, these are what we call the construction lines, lines for sketches.

Draw a square, use diagonals to find the center,

and gradually construct the arcs and form a circle. The second method which we can use, especially when we are drawing a bigger circle, is that,

when we draw a square, if we only rely on these four points to construct the arcs, it is relatively harder. A simpler way,

what I call Isometric-assisted method, we use the property of center and radius and we can construct a complete circle.

I know where the center is, I can take a small strip of paper and based one end at the center, I rotate this strip of paper,

then, I will know where every point of my circle should be.

So I plot these dots and draw the arcs to form a circle.

To draw a small circle, this may not be necessary, the Square method will do.

But for bigger circles, for examples, for posters, to draw such big circles,

we will need more dots for our reference, then, it will be advisable to use a strip of paper, or maybe a ruler,

or any object that can give you a fixed distance, and so you can plot the dots for reference,

and then connect the dots with arcs to form circles.

The effect of such a method is quite good. The third common method for drawing circle freehand is the Pivot method.

You can understand that when engineers are drawing,

the little finger is quite useless as we hold the pencil in our hand. We can use that little finger as pivot, as the center of a circle.

Well, you will discover that, this way, for a right-handed person like me, it is very easy to draw from bottom right corner, to top left corner.

It is very easy to plot such a arc. Then, I will have to rotate the paper.

For as long as my little finger is at the center, and I draw with this method, I will be able to draw a medium size circle.

You can imagine what size of a circle my hand is able to draw. With this method, the circle we construct will just be about this size,

a circle that we usually fit in our papers. If you want a circle that is very huge or very tiny, this method will not be suitable.

But in engineering, when we draw freehand, we don't usually want very huge circles or very tiny circles, so this method will do.

In the Pivot method, another important point is the rotation of paper.

the smooth movement of our wrist is quite limited to certain angles. It's not easy to be smooth in the whole movement.

That is why we have to rotate the paper to draw a smooth circle. In the pivot method, we can also use two pencils instead.

So, a pencil is for the reference of the center,

while the other is for construction of the circle. We will be able to plot a circle like that.

Ellipses are quite similar to circles.

An ellipse is a circle with different width and length.

You can imagine ellipse as a squashed circle.

So, when we are drawing an ellipse, we can squash the Square method to a Rectangle method.

First, draw a rectangle, then, draw the diagonals.

Then, we find the center of the ellipse.

After that, we can develop this ellipse shape with the junctures of the rectangle and ellipse,

there are junctures at the top and bottom, also on the left and on the right.

Well, we have to use faint lines to construct the ellipse.

Then, we trace it with solid lines, this will give us very good effects.

So, the Rectangle method is that we draw a rectangle with faint lines,

find the center with diagonals and draw the ellipse.

Then, arc, is part of a circle, so we just need to find its circular properties,

then, we will be able to draw arcs nicely. As you can see here, if we are to draw a arc,

we have to draw the relevant construction lines. Then, we can draw the arc the same way we draw circles.

So, as a simple conclusion, the construction of circles is done by mostly three methods:

Square method, Isometric-assisted method and Pivot method. Well, we have covered them all today. I hope you will practice them.

Then, for ellipses, the transformation of circles, you change Square to Rectangles.

then, you will draw the ellipse in almost the same way as you draw circles.

Let the arcs extend and form the ellipse.

Lastly, arc is part of a circle. Here we did not emphasis a lot on this.

However, since it is part of a circle, all we need to do, is to find its center and radius, and we can plot the arc the same way we draw circles.

Therefore, if you can draw circles,

Ellipses and arcs should not be a problem. So far, we have talked about lines,

and we also talked about circles, ellipses and arcs. In fact, Engineering Graphics is composed of these few basic elements,