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I am really confused regarding this topic. Suppose I want to draw a cone. There are two ways to draw it, upside down and the standard upside up. Now as a viewer from above, how would I draw these figures on a 2D paper. To me both diagram look same. Note shading is not allowed, only contour lines!

Essentially what I am asking is this:

I recall that to show the illusion of depth, we let parallel lines coverage. My question is the converse, how should the parallel lines behave when I want to show the illusion of height/protrusion.

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  • You're talking about "both diagrams": it would be helpful to edit in images. "Upside down" and "standard upside up" don't really bring anything particular to my mind. Contour lines are the lines that enclose an object from one's point of view: are you asking about lines that are (always) parallel to those, so parallel to one's picture plane?
    – Joachim
    Commented Jan 31, 2023 at 15:05
  • Should "we let parallel lines coverage" be "we let parallel lines converge"?
    – Joachim
    Commented Feb 1, 2023 at 16:03

3 Answers 3

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If it's specifically a topographic map, then for depressions you use contours with small lines pointing towards holes

Image of the countour

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There are a couple of basic principles that define how contour lines behave. Let me start with that.

Say you hold a ruler vertically in front of your face. You see the full height of the ruler:

enter image description here

If you tilt the ruler away from you, it appears shorter because some of its length is used in a direction away from you, which doesn't affect it's vertical appearance:

enter image description here

If you tilted it to horizontal, it would disappear (well, except for the physical end of the ruler, which has some thickness).

The ruler has a flat surface. If you tilt it away from you, not just the ruler's length gets shorter. The markings also get shorter (closer together). But the spacing between them stays uniform. The marking are still evenly spaced, just less distance between them because they all fit in a reduced visual height.

A curved surface is different. For this example, think of a globe of the earth. Imagine we were to slice the globe in half, cutting from the north pole to the south pole, and then looked at the cut face. And imagine that a clock face is superimposed on that, with the north pole being 12:00 and the south pole 6:00. The hours would divide the edge of the globe's surface into equal lengths (the distance around the cut edge of the globe would be the same between every hour).

We mark the hour locations and put the globe back together. Then we create a "latitude" line running through 1:00 and 11:00, another latitude line running through 2:00 and 10:00, and the equator runs through 3:00 and 9:00:

enter image description here

I did that by hand on a spreadsheet, so it isn't perfectly to scale, but it shows the idea (ignore that the remaining pictures are even worse).

If you look down at the globe, directly at the north pole, the curvature affects what you see. The curvature doesn't have a big effect on the view down to the 1:00-11:00 latitude; not much of the surface is in a direction away from you. Between the 2:00-10:00 latitude and the equator, the surface is oriented in a direction mostly away from you, so little of it affects the dimensions you can see:

enter image description here

Remember, each of those latitudes is the same distance apart on the surface, but your view looking down at the north pole is this:

enter image description here

I warned you the drawings get worse. The contour lines aren't evenly spaced, but the spacing isn't random (that point might be hard to see with just the couple of lines in the diagram). The spacing follows a progression of closer to farther apart based on the curvature. On a complex surface, like say a face, the changing directions of the features will be reflected in the contour lines changing the distance between them and changing the progression of separation distance.

So the way contour lines work is that they represent lines an equal distance apart along the actual surface. If the surface is flat, the lines will be evenly spaced. If it is curved, the spacing will follow a progression. If the surface is tilted or curved perpendicular to your view, the lines will appear closer together. The steeper the tilt, the closer the lines will be.

So how does it work with a cone? A cone is both curved and flat. If the cone is vertical, you can't draw verticalish contour lines (at least in the most intuitive way), because the surface goes from a wide base to a point; lines on the surface wouldn't be parallel, they would converge at the point. The most intuitive way to get parallel lines is to run them in the circular direction:

enter image description here

In the height direction, the cone behaves like a flat surface, the lines remain equally spaced. It doesn't matter whether the cone is right-side-up or up-side-down. But the portion of the contour lines facing you follow the same shape as the base, the circumference just gets smaller as you go up the cone. The contour lines show the shape of the cone by their curvature, similar to the base, and the equal spacing between them.

As Matt's answer points out, you can create parallel lines in other directions on the surface of a cone (even vertical), e.g., the single axis "worm" diagram in that answer. That works just as well. It can be a little less intuitive to create as a first pass at getting the hang of contour lines, so I didn't cover it in this answer, but think of that as "phase 2" of contour lines.

That answer also makes another important point not to miss. The contour line spacing is affected by the surface's orientation perpendicular to you, but that orientation doesn't have a direction (whether you move towards you or away from you along that orientation doesn't affect the contour line spacing). Many optical illusions are based on this ambiguity. In creating contour lines, you can ignore whether the surface is protruding or receding.

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There are two things you might be talking about. In topography and other formal diagramming systems, there are conventional standards for indicating elevation changes. Affaltar provides some information about that in their answer. If you're interested in diagramming, it would be helpful to know what kind of diagramming, and which discipline you're working in.

The more "artistic" exercise of drawing contours has less to do with execution, and more to do with training your brain. The practice is an exercise in picturing an object in your mind, associating that object with the outline on the page, and then imagining what lines would look like if they were wrapped around it.

The result is that your brain learns to recognize the contours you've drawn, as much as it learns to produce the contours you will draw. This practice of drawing is more about holding a real object in your mind, and representing reality, than it is about providing information about the shape of the object. It's easy for REAL contour lines around a REAL object to produce optical illusions that make it hard to tell what shape the object is. So, if your contours look "wrong" or odd, one reason might be that it's one of those real situations where the contours in real life produce an optical illusion, or are difficult to "read." Another reason, is that your brain is still learning to interpret the lines you've drawn. They might BE correct, but LOOK wrong.

Now, that's a very "artsy-fartsy" answer, so to be a bit more practical: don't forget that your contours should be around a single axis, and they should be the same distance from each other on whatever axis you've chosen to draw them around. Drawing your contours in perspective is also a good way to help the shape "read" more easily. If you're drawing contours in perspective, things can get a little complicated, which is where the brain-training comes into play. Once I get a minute, I'll see if I can upload some examples.

I'll reiterate: This is for drawing that is not interested in being restricted to a particular diagramming paradigm. If you're interested in diagramming, your solutions will be related to the conventions of that particular type of diagramming (e.g. architectural blueprints, engineering designs, topographical maps, etc.). Similarly, if you're wanting to draw a cylinder, directly from the top, orthographically, you're going to have a hard time making it look like anything more than concentric circles.

Aside from those qualifiers, here are some tips and tricks to help make contour drawing a bit more straightforward.

Optical Illusions:

Here's an example of an optical illusion I mentioned.

Optical Illusion

In this case, we can't tell if the shape in the plane is a bump or a dimple, because there's just not enough information drawn. It's not wrong, but if we add some information about where the viewer's eye is relative to the plane, it will "read" more easily. One way to add such information is with a horizon line, or by drawing in the thickness of the plane's edges, which will give our eyes a clue about where the perspective lines are converging to.

Single Axis:

Sometimes, it's tempting to draw contours that following the "cylindrical" shapes of an object, because they're convenient. This is certainly a valid style, and has its place, but as an analytical exercise for practicing, it's not as useful.

Not Single Axis

Studying the 3D shape of an object is often more useful when restricting ourselves to a single axis (even if it's not a conventional vertical or horizontal axis).

Single Axis

Evenly Spaced:

A slightly more exaggerated version of the bump/dimple on a plane. You can see that our contours are made at even intervals, but the curves that trace the bump/dimple end up getting close to each other because of how perspective distorts them. Making the contours an even distance will allow the distortions due to perspective to sell the shape better. Uneven "cuts" make it hard to tell whether the distortions are due to perspective, or irregular spacing.

Evenly Spaced

Some details to pay attention to that will help sell the dimensionality of your contours are:

Drawing in perspective:

Think of your contours flat discs in space. Their shape will change depending on whether they are above the viewer's eye level, or below it. Thinking of discs that are stacked like a column right in front of you, just a vertical pillar: one of the discs far above the viewer's eye level will appear very round. The same disc close to the viewer's eye level, right in front of them, will look like just a straight line. The same is true with your contours. A contour that's around a part of the object far above the viewer will look more like its cross-section. A contour that's closer to the viewer's eye level will look more like a straight line.

For example, with this cylinder drawn orthographically, the curves are all the same shape, because they're not drawn in perspective.

Orthographic

With this cylinder drawn in perspective, the curves that are farther below the eyeline have a more extreme curve to them.

Perspective

Perspective Occlusion:

When contours are receding away from the viewer in the right way, one or both ends of the contour may be occluded by the sides of the object. In terms of "line" the contour will simply merge with the silhouette, instead of showing a slight bend.

On parts of the shape that are closer, we may see a slight bend at the end of the contour line as it begins to wrap around the object. We may even see the whole contour altogether.

Perspective Occlusion

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