In 1692, Dutch artist A. Boogert made a one-of-a-kind teaching manual with over 800 pages of color swatches, meticulously cataloguing the many possible colors that he could mix from watercolors.

More at This is Collossal

Thanks to Roberto Quintano, Dustin Wilson, Tim Fehr, and Steve Gilzow for telling me about it.

## 20 comments:

Hi James,

Sorry for the comment unrelated to the post. I have a question about perspective. In Andrew Loomis' "Successful Drawing," he points out the connection between shading and perspective by showing that, for example, the shade line on a sphere is found via perspective. Unfortunately, though he shows this in diagrams, he doesn't explain how to do it. I've tried to solve the problem on my own without success, and I've done a lot of searching for an answer. It seems that this problem was addressed in many old books about architectural perspective, but all the books I found online either had missing diagrams or referred to non-perspective architectural projections. My question is: Do you know of a straightforward method for finding the shade line and cast shadow of a sphere in perspective, or perhaps a book that explains it?

Thanks!

Dan

Dan, the basic thing to keep in mind is that the circle where the shadow forms on a sphere is where the light rays are tangent to the edge of the sphere. Assuming the light rays are parellel, the circle is also the great circle formed through the axis through the center that a light ray would take. Best way to see this is by holding up an orange in the sun and studying the shadow.

James: Thanks! I gather that the ellipse for the great circle in perspective will have the same center as the sphere, will have a major axis equal to the diameter of the sphere, and perpendicular to the light ray through the center of the sphere, and will have a minor axis along this light ray centered at the center point. The only problem is: how to determine the length of the minor axis. Do you know of a straightforward way?

Thanks again!

Dan

this is amazing. thanks for sharing this.

Ah, so that's where Lee, Rosco and Gam got their idea from (just kidding; of course this is not additive). I wonder if he had a CIE chromaticity scale, too :)

Dan~ Now don't laugh but there is a simple and very good example of cast shadows in the Walter Foster book on perspective. It show how to figure direct and indirect light sources. You should be able to find one at most art stores or online through ebay or a google search.

Kevin: Thanks. No laughing here. I've seen lots of basics on cast shadows, especially of forms like cubes, cones, and cylinders, as well as even complex forms.

But the sphere seems to somehow be in a class by itself when it comes to most of perspective problems, because it doesn't have edges or many easily locatable points. For example, it's even a difficult perspective problem to take an existing drawing of a sphere and enclose it in a cube. (It's somewhat easier to construct one inside an existing cube, but by no means simple.)

I'm new to all this, but starting to wonder whether all artists don't simply "wing it" with spherical shapes, going mostly on what they have observed in the past when constructing a scene, or else setting up a model of some kind.

Or maybe I'm way off base and there are a lot of simple rules governing the shades and shadows of the sphere that I simply haven't found yet. At least not in Norling, Cole, Loomis, et al.

I'm trying to learn art, not mechanical drafting. I'd just like to gain a thorough understanding of the principles of perspective, and when I saw that tantalizing idea in Loomis' book about the perspective of the shade line of a sphere, I started searching for a perspective solution to that problem. If someone in the know wants to chime in with, "Actually there isn't one," I'll be happy to accept that! :)

Thanks,

Dan

Apologies James~

Dan, I have a quick sketch to illustrate how you can do the shadow cast by a sphere in perspective.

www.kevinmayes.com/sphere

It's not as difficult as you might think!

Kevin and Dan, thanks for helping each other figure this out. I've been swamped with another project, but I appreciate the exchange you're doing--I'm sure you're helping others, too.

@ Dan - First, yes the sphere is in a class by itself when it comes to perspective. There really is no foreshortening taking place as with other forms until you slice into it then ellipse' describe that foreshortening.

There probably is a way to plot out the perspective of the minor axis of the shadow line although I'll bet it is gets complicated because it is determined by the point of view (the viewer) in relation the light source.

If the shadow line is a plane (a slice of the sphere) there must be a way.

Now I am curious too.

I checked Handprint.com not it did not address your question specifically.

http://www.handprint.com/HP/WCL/perspect5.html

Hi Kevin,

The two points where the light is tangent to the circle are the ends of the major axis of the shade line ellipse.

And the ends of the minor axis fall along a line from the light source through the center of the circle. But how do you determine which two points they are on this line? They are centered at the center (ignoring for now the difficulty that the center of the ellipse will not fall exactly at the center of the circle due to foreshortening).

The same answer is needed to solve the cast shadow. Its major axis is found where the rays tangent to the circle intersect a line from the base of the sphere to the shadow VP. Its minor axis requires the shade line ellipse to be known first. It seems to be bounded where rays tangent to the ellipse of the shade line intersect the perpendicular to the major axis. Yes?

The length of that minor axis of the shade line is the main problem. I suppose one could figure out the exact angle of the ellipse with respect to the viewer and compute the cosine of the angle multiplied by the major axis. I was wondering whether there is a straightforward, geometric "perspective" way to do it.

The diagram you drew looks correct, and it is very like the one in Loomis. Did you just "eyeball" the width of the ellipse, or did I miss some important point?

Thanks!

Dan

Oh, and David: Thanks for your help! The shadow line is where the plane through the center of the sphere, perpendicular to the ray of light through the center of the sphere, intersects the sphere. If a perspective square could be drawn on this plane that bounds the edges of the ellipse, then the ellipse could be drawn within this square. I think that could be done if you worked at it hard enough. You could turn the drawing around the principal VP so that the light rays appear to be level, then find the VP for perpendiculars to light rays based on a station point. I started down this path, but it quickly became unweildy. It seemed to me that there must be some easier way in practice.

James: Thanks for the forum!

Regards,

Dan

Dan~

I have done this long enough to 'eyeball' the width, but to be accurate draw a bounding box for the sphere. The center of each plane would be where the sphere touches. The surface could be gridded to the vanishing points to give a surface to project onto. The length of the major axis for the shadow is relative to the light source. Project a line from the light source to the upper most point of the sphere to the surface to establish your shadows furthest point. Place points on the ellipse around the sphere and project from the light source to the surface through those points. Now connect the dots on the surface plane and you have your shadow in perspective!

Hope this helps. If you want you can email me through my site.

@ Dan - Just to be clear here, the ellipse you are try to determine is the core shadow ellipse (shadow line) right?

That is my interpretation of your question.

It's the core shadow line that gives the sphere volume and depending on point of view could be a concave or convex arc across the sphere.

The cast shadow is a bit easier to determine.

David, yes that's right. I'm looking for the ellipse that describes the great circle dividing the sphere into its light and shadow halves. The idea is to find a way to construct this line in perspective given the sphere and the direction of light (assuming it's sunlight).

It might be a matter of treating the plane of the ellipse as a slope.

If the direction of light, and plane of the ellipse being perpendicular to it, establishes the angle of slope.

and

If we know the length of major axis (the diameter of sphere) then using the rules of perspective we can transfer that length (move it around using measuring points) to establish the length of the minor axis, since an ellipse is a circle in perspective and both axis' would be the same (length) in geometry.

Unless that is what you refer to as unwieldy in earlier comment.

?

I'll have to try it.

Thanks James, hope you don't mind us highjacking the post here!

David, I had that same basic idea, but I've found it difficult to work out in practice. Will be very interested to see what you come up with.

Thanks,

Dan

Well Dan,

I have had no success so far at finding a straightforward or simpler way to finding the shade line especially and the cast shadow of a sphere in perspective.

The sphere is unique and it gets complicated fast.

There must be a way and I'll keep working on it just because I too want to know now.

CURSE THE SPHERE!

That's what is called research work.

This artist was a real scientist.

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