HOME | DD

Description
This is about about something a bit different: how to draw specific kinds of triangles.

So far, most of this tutorial has been either about squares and rectangles, or curved lines (4VP onward). But even those were squares and rectangles, if somewhat warped and abstract.

But what about other shapes?

There have been parts here about drawing pyramids, but their bases have always been squares. What if you want to draw a triangle at an angle? Say, for a three-sided pyramid?

"But wait- I can already! Once you have a square or rectangle, all you have to do is draw a line connecting two opposite corners, and you have your triangle- two of them, in fact!"

Well, that is true- but you'll always end up with "right" triangles, that is, triangles with a 90 degree angle in them. What if you want an equilateral triangle, a triangle with all sides the same and all three inside angles 60 degrees? An isosceles triangle, one with two equal sides?

(Hint: The three inside angles of a triangle always equal 180 degrees)

That's what this section is about- how to draw specific kinds of triangles!

Fig. 1 shows a trick we learned back in the 1970s for drawing an equilateral triangle- you draw your base line (Line AB), then take a compass and put its ends on points A and B. Swing it up as shown- then switch the compass points around, and do it again. Where those arcs meet is where the third corner of the triangle will go.

It is directly above the midpoint of Line AB and, since the compass ends were on points A and B, that third point must be the same distance from points A and B as A and B are from each other.

You have your equilateral triangle.

(Isosceles triangle- instead of putting both ends of the compass on A and B, make it go past or not as far as the opposite point for the first arc, then just switch the ends around and make your second arc. It'll work the same way)

Fig. 2 has the triangle, but here is where it's easy to make a common mistake: an equilateral triangle CANNOT be drawn inside of a square! The dotted line on the left side is as long as Line AB- see how it is taller than the triangle? To enclose the triangle, you must have a rectangle that is shorter than a square...

And so, Fig. 3. The apex and base of the triangle touch the longer sides here.

***********
And this is an important thing to understand- not the image itself, but the concept behind it. Figs. 1, 2, and 3 are all basic non-angled images, yet they are what let me figure this part out. It was the same with the "Even Width Arches," too. There will certainly be things you will want to do that are NOT specifically covered in this or maybe any other tutorial you may find, so you might have to try to figure it out. By starting out like this you can see clearly what exactly you are up against and at least have a basic idea how it works. And that can take you a lot closer than you might think.
**********

Fig. 4 has two things we need to know: corner angles and length of sides. A rectangle will have two different corner angles and a short side and a longer side. To do the rest of this, we need to know what they are. And so- the angles and lengths!

As with Fig. 5. This is simply the rectangle (and the triangle) turned on its side. The shape itself has not changed, but now it is taller than before. Compare it to Fig. 4.

And now, for the tricky part...

At this point, if you are new to this tutorial, you really should check out these pages:

[link]

[link]

Those two pages, from the earliest part of the whole tutorial, are THE basis for what's being done here.

Fig. 6 shows a typical One Vanishing Point Horizon, with the Vanishing Point on it. The lower side of the surface is where that point- the one telling you not to ignore it!- is.

Thus, to get a square, you draw a point exactly as far left or right (or both) from the VP as that low point is below the VP- in this case, 10 spaces. Those are Points A and B, and they are both 10 spaces away from VP.

If you draw a horizontal line somewhere, connect its ends to the VP- again, typical 1VP style- and then from either end of the horizontal line draw a line to A or B- whichever will make it cut through the lines from the VP- you'll end up with a square.

But we know we do NOT want a square; we DO want a rectangle!

To get one, you'd move Point A to the left or right. Likewise, Point B.

But how far?

Math. At times I'm going to round off just a bit, by the way.

In Fig. 4, divide 4.5 / 5.1. This gives you 0.9; 9/10th; 90%. The side is 1/10 (10%) SHORTER than the base.

For Fig. 5, divide 5.1 / 4.5. This will give you 1.1; 11/10 (1 1/10); 110%. The side is 1/10 LONGER than the base.

You're ready for the tricky part.

To keep it easy, I made sure that Points A and B were 10 spaces from VP, as was that lower point.

You've noticed that I use fractions here. That is because of the way I prefer to do it, so if I get, say, 1.10 (or 110%) you know that is 1 1/10; if you get 0.9 (90%), you know that is 9/10. I usually use fractions, but decimals can be used, too. The set-up should be the same, though.

And here's why: to get a TALLER-than-square shape, you must use a point that is CLOSER to the VP than Point A or B.

If you want a SHORTER-than-square shape, you must use a point that is FURTHER AWAY from the VP than Point A or B.

Closer=taller, and further away=shorter. It's "inverse."

And so, you flip the fractions (or decimals...).

In the case of Fig. 4, it is 9/10 as tall. Flip it to 10/9, and you get 1 1/9. That is 1 (100%, or all ten spaces here) and 1/9 (1 divided by 9, which equals 1 (rounded off).) 10+1=11, which is how far Point C and Point E happen to be from VP.

For Fig. 5, it is 1 1/10 as tall. 1 1/10 is 11/10. Flip it to 10/11. 10 divided by 11 equals 0.9 (rounded off). That's close to 9/10, or 9 spaces, which is how far Point D and Point F happen to be from VP.

Of course, you can use decimals if you like. You can, for Fig. 4, use 4.5/5.1 instead of a regular fraction; this will also give you 0.9. You can, just as with the fraction, flip it to 5.1/4.5; this will also give 1.1. Multiply that times 10 spaces and you get 10 X 1.1- which equals 11 spaces!

Fig. 5 would be 5.1/4.5=1.1; flip it to 4.5/5.1=0.9...and 0.9 X 10= 9 spaces!

If you had 12 spaces instead of 10, then it would be 12 X 1.1= 13.2 spaces. 12 X 0.9= 10.8 spaces.

With those new points, you can easily draw your rectangles as shown in Fig. 6! You've gotten through the tough part!

SUPPLEMENT: A BIT ON FRACTIONS...

When dealing with Fractions, Decimals, and Percentages, it helps to have a reference, something so you can compare them all.

With Percentages, 100% means "everything." 50% means "half of everything."

In Decimals, this would be equal to 1.00 and 0.50. (You can convert decimals to percentages just by moving the period two spaces to the right, so 0.50 becomes 50%. Move it left two spaces to convert the percentage back to a decimal.)

With Fractions, this would be equal to 1 and 1/2, although it could also be 100 and 50/100; it could be 50 and 25/50...Fractions can be literal, so 50/100 can literally mean "50 out of 100."

What may be confusing is something like 11/10, or 1 1/10. Those two fractions are the same exact thing. Here's how:

1/10; the top number (1) is the "Numerator," while the lower number (10) is the "Denominator." This is true for all fractions, no matter what the top and bottom numbers are.

With 1 1/10, you have a whole number (that first 1) and a fraction (1/10) together. This whole thing means at least more than 100%.

To convert it into a simple typical-style fraction, multiply the Denominator X the Whole Number; then add the Numerator to that. So...

10 X 1= 10; 10+1=11!

Put the result over the Denominator, and you have it: 11/10!

To break down 11/10 into a fancier complex fraction, first divide the Numerator by the Denominator. In this case, it would be 11 divided by 10, but at this point ONLY GO AS FAR AS GETTING A WHOLE NUMBER.

In other words, 11/10 = 1. What's left? Well, 10 goes into 11 once (1), and you have 1 left over- it's "1 remainder 1."

Right!

Put the whole number first, then put the remainder over the Denominator. That would be 1 1/10.

O.k., how about, say, 2 4/5?

That's (5X2=10)+(4). That's 14. Put it over the 5. "14/5."

14/5, break it down? 5 goes into 14 twice (2), with 4 left over. That's...2 4/5.

You can even be nasty about it: take 3 4/3!

3 4/3. What exactly is it? Well, you have 3. You also know that 4/3 is actually 1 1/3; 4 divided by 3 = 1 remainder 1.

So it's 3 + 1 1/3, or 4 1/3.

Let's try it directly:

3 4/3 = (3 X 3) + (4) = 13/3. Which happens to be 4 1/3. Same thing.

Now don't let anyone tell you art isn't educational!

Description
This is about about something a bit different: how to draw specific kinds of triangles.

So far, most of this tutorial has been either about squares and rectangles, or curved lines (4VP onward). But even those were squares and rectangles, if somewhat warped and abstract.

But what about other shapes?

There have been parts here about drawing pyramids, but their bases have always been squares. What if you want to draw a triangle at an angle? Say, for a three-sided pyramid?

"But wait- I can already! Once you have a square or rectangle, all you have to do is draw a line connecting two opposite corners, and you have your triangle- two of them, in fact!"

Well, that is true- but you'll always end up with "right" triangles, that is, triangles with a 90 degree angle in them. What if you want an equilateral triangle, a triangle with all sides the same and all three inside angles 60 degrees? An isosceles triangle, one with two equal sides?

(Hint: The three inside angles of a triangle always equal 180 degrees)

That's what this section is about- how to draw specific kinds of triangles!

Fig. 1 shows a trick we learned back in the 1970s for drawing an equilateral triangle- you draw your base line (Line AB), then take a compass and put its ends on points A and B. Swing it up as shown- then switch the compass points around, and do it again. Where those arcs meet is where the third corner of the triangle will go.

It is directly above the midpoint of Line AB and, since the compass ends were on points A and B, that third point must be the same distance from points A and B as A and B are from each other.

You have your equilateral triangle.

(Isosceles triangle- instead of putting both ends of the compass on A and B, make it go past or not as far as the opposite point for the first arc, then just switch the ends around and make your second arc. It'll work the same way)

Fig. 2 has the triangle, but here is where it's easy to make a common mistake: an equilateral triangle CANNOT be drawn inside of a square! The dotted line on the left side is as long as Line AB- see how it is taller than the triangle? To enclose the triangle, you must have a rectangle that is shorter than a square...

And so, Fig. 3. The apex and base of the triangle touch the longer sides here.

***********
And this is an important thing to understand- not the image itself, but the concept behind it. Figs. 1, 2, and 3 are all basic non-angled images, yet they are what let me figure this part out. It was the same with the "Even Width Arches," too. There will certainly be things you will want to do that are NOT specifically covered in this or maybe any other tutorial you may find, so you might have to try to figure it out. By starting out like this you can see clearly what exactly you are up against and at least have a basic idea how it works. And that can take you a lot closer than you might think.
**********

Fig. 4 has two things we need to know: corner angles and length of sides. A rectangle will have two different corner angles and a short side and a longer side. To do the rest of this, we need to know what they are. And so- the angles and lengths!

As with Fig. 5. This is simply the rectangle (and the triangle) turned on its side. The shape itself has not changed, but now it is taller than before. Compare it to Fig. 4.

And now, for the tricky part...

At this point, if you are new to this tutorial, you really should check out these pages:

[link]

[link]

Those two pages, from the earliest part of the whole tutorial, are THE basis for what's being done here.

Fig. 6 shows a typical One Vanishing Point Horizon, with the Vanishing Point on it. The lower side of the surface is where that point- the one telling you not to ignore it!- is.

Thus, to get a square, you draw a point exactly as far left or right (or both) from the VP as that low point is below the VP- in this case, 10 spaces. Those are Points A and B, and they are both 10 spaces away from VP.

If you draw a horizontal line somewhere, connect its ends to the VP- again, typical 1VP style- and then from either end of the horizontal line draw a line to A or B- whichever will make it cut through the lines from the VP- you'll end up with a square.

But we know we do NOT want a square; we DO want a rectangle!

To get one, you'd move Point A to the left or right. Likewise, Point B.

But how far?

Math. At times I'm going to round off just a bit, by the way.

In Fig. 4, divide 4.5 / 5.1. This gives you 0.9; 9/10th; 90%. The side is 1/10 (10%) SHORTER than the base.

For Fig. 5, divide 5.1 / 4.5. This will give you 1.1; 11/10 (1 1/10); 110%. The side is 1/10 LONGER than the base.

You're ready for the tricky part.

To keep it easy, I made sure that Points A and B were 10 spaces from VP, as was that lower point.

You've noticed that I use fractions here. That is because of the way I prefer to do it, so if I get, say, 1.10 (or 110%) you know that is 1 1/10; if you get 0.9 (90%), you know that is 9/10. I usually use fractions, but decimals can be used, too. The set-up should be the same, though.

And here's why: to get a TALLER-than-square shape, you must use a point that is CLOSER to the VP than Point A or B.

If you want a SHORTER-than-square shape, you must use a point that is FURTHER AWAY from the VP than Point A or B.

Closer=taller, and further away=shorter. It's "inverse."

And so, you flip the fractions (or decimals...).

In the case of Fig. 4, it is 9/10 as tall. Flip it to 10/9, and you get 1 1/9. That is 1 (100%, or all ten spaces here) and 1/9 (1 divided by 9, which equals 1 (rounded off).) 10+1=11, which is how far Point C and Point E happen to be from VP.

For Fig. 5, it is 1 1/10 as tall. 1 1/10 is 11/10. Flip it to 10/11. 10 divided by 11 equals 0.9 (rounded off). That's close to 9/10, or 9 spaces, which is how far Point D and Point F happen to be from VP.

Of course, you can use decimals if you like. You can, for Fig. 4, use 4.5/5.1 instead of a regular fraction; this will also give you 0.9. You can, just as with the fraction, flip it to 5.1/4.5; this will also give 1.1. Multiply that times 10 spaces and you get 10 X 1.1- which equals 11 spaces!

Fig. 5 would be 5.1/4.5=1.1; flip it to 4.5/5.1=0.9...and 0.9 X 10= 9 spaces!

If you had 12 spaces instead of 10, then it would be 12 X 1.1= 13.2 spaces. 12 X 0.9= 10.8 spaces.

With those new points, you can easily draw your rectangles as shown in Fig. 6! You've gotten through the tough part!

SUPPLEMENT: A BIT ON FRACTIONS...

When dealing with Fractions, Decimals, and Percentages, it helps to have a reference, something so you can compare them all.

With Percentages, 100% means "everything." 50% means "half of everything."

In Decimals, this would be equal to 1.00 and 0.50. (You can convert decimals to percentages just by moving the period two spaces to the right, so 0.50 becomes 50%. Move it left two spaces to convert the percentage back to a decimal.)

With Fractions, this would be equal to 1 and 1/2, although it could also be 100 and 50/100; it could be 50 and 25/50...Fractions can be literal, so 50/100 can literally mean "50 out of 100."

What may be confusing is something like 11/10, or 1 1/10. Those two fractions are the same exact thing. Here's how:

1/10; the top number (1) is the "Numerator," while the lower number (10) is the "Denominator." This is true for all fractions, no matter what the top and bottom numbers are.

With 1 1/10, you have a whole number (that first 1) and a fraction (1/10) together. This whole thing means at least more than 100%.

To convert it into a simple typical-style fraction, multiply the Denominator X the Whole Number; then add the Numerator to that. So...

10 X 1= 10; 10+1=11!

Put the result over the Denominator, and you have it: 11/10!

To break down 11/10 into a fancier complex fraction, first divide the Numerator by the Denominator. In this case, it would be 11 divided by 10, but at this point ONLY GO AS FAR AS GETTING A WHOLE NUMBER.

In other words, 11/10 = 1. What's left? Well, 10 goes into 11 once (1), and you have 1 left over- it's "1 remainder 1."

Right!

Put the whole number first, then put the remainder over the Denominator. That would be 1 1/10.

O.k., how about, say, 2 4/5?

That's (5X2=10)+(4). That's 14. Put it over the 5. "14/5."

14/5, break it down? 5 goes into 14 twice (2), with 4 left over. That's...2 4/5.

You can even be nasty about it: take 3 4/3!

3 4/3. What exactly is it? Well, you have 3. You also know that 4/3 is actually 1 1/3; 4 divided by 3 = 1 remainder 1.

So it's 3 + 1 1/3, or 4 1/3.

Let's try it directly:

3 4/3 = (3 X 3) + (4) = 13/3. Which happens to be 4 1/3. Same thing.

Now don't let anyone tell you art isn't educational!