The method of placing a shadow into perspective is an important skill to have as an illustrator.  There is a method which Andrew Loomis shows us in his book called Successful Drawing.  I will outline it here, following a short preamble.

When we look at an object on a plane, we see it in perspective, as it appears from our point of view-or our perspective.  The angles recede to a point on the horizon called the Vanishing Point, which always changes depending on where the viewer is situated.

The shadows cast by an object sitting on a plane will also have a vanishing point, but it is relative to the direction the light is being cast, as well as to the viewer’s point of view.  Working out the perspective the shadow takes, relative to the viewer is a factor of drawing the primary object into proper perspective- if you have done this, the P.O.V perspective of the object’s shadow will follow suit.  This is only the first consideration though, as stated above, the shadow also has to fall relative to the direction the light is being cast.

I will demonstrate Loomis’ method of solving this problem.

1. First, as will all drawing, one should set out the horizon line, and decide on the vanishing point, or points if the perspective is 2 point or more.  Here is a sample with 2 Vanishing Points, labeled V.P. below.
2. Once the VPs are set up, draw lines going to them, as per the shape you are drawing.  It is important to draw a geometric shape at this stage, then convert it to an organic shape later once the perspective is worked out, since it is easier to solve these issues geometrically.   Here I intend to draw a box shape.  
3. Here is the box shape drawn in using the two VPs.
4. Now, once you have decided on the object’s perspective, it is time to decide on the next major piece of information which an illustrator needs, in order to create a convincing picture; the direction of light.  You should be aware that this stage is not regarding the angle the light is falling from, just whether it is coming from the left of the object or the right.  The angle is determined later.  For this image, I chose the light to be coming from behind-right. (NB. Last posting defined the direction terms Behind and Before as being relative to the viewer, not the subject matter.  If the light is behind the viewer, the direction line comes from below.)
5. Light travels in a straight line, so this direction line you set is very important; it will now show us where the VP of the shadow will be, relative to the light.  As stated before, the shadow’s perspective relative to you will be taken care of on account of the shape you have drawn already.  Whichever perspective that shape is in will define the POV perspective the shadow takes, as we will use the corners of the shape as reference points for the shadow.  When you cast the line the light is travelling at, the point where it crosses the horizon will be the VP of the shadow.  Bisect the horizon with a vertical line where the Shadow VP lays.
6. Once the vertical line bisecting the Shadow VP has been established, we can now determine the angle of the light.  The vertical drawn will act as our gauge to keep the angle constant.  Choose the most prominent corner of your subject matter, and at an angle of your choosing, cast a line which goes through the prominent corner, as well as through the vertical line you drew in step 5. This will be the primary angle of the light.
7. Since there are more than 1 corner which catch the falling light, we need to determine where that far, and hidden corner of the box would be.  This is done by again casting lines to the 2 original VPs we chose for the box.  Cast your lines to determine the back side of the box using the corners which you can see on the right and left.  I used the blue lines to demonstrate this below.
8. Once that is done, describe the back side of the box, using dotted lines, and make a note of where the corners are.
9. The second upper corner of the box, hidden by the side which faces you is also within the purview of the light which is falling.  You need to now account for the angle of the shadow it casts, using the first Angle of Light you decided upon.  At the crossing point of the Angle of Light and the vertical line which bisects the Shadow VP, draw up again, crossing through the new, hidden corner you just solved in step 8.  This will tell you how the shadow falls relative to that corner, and the angle of light.   This new angle line I have drawn below, it is the thinner of the two lines describing the Angle of Light.
10. Now, using the heretofore unmentioned bottom corners, cast lines from both the near and far corners, going to the Shadow VP.  These lines will give us the side edges of the cast shadow, and they are represented in dotted blue below.  The red Angle of Light lines show us where the shadow will end, in a straight line between the two crossing points of the 2 dotted blues, with the two reds.
11. Please note, I marked an X where the far corner of the shadow would appear (at the crossing point of dotted blue and the thin red Angle of light line).  You can see that the X is within the body of the subject matter, indicating that the back corner of the shadow cannot be perceived by a viewer standing at the perspective we chose for this object.  This is what I was meaning by saying that by properly drawing the subject matter in perspective will take care of the shadow’s P.O.V perspective.  Here is the X marked in the drawing below, with pink lines outlining the box’s shadow which such a perspective and angle of light would cast.  The far corner of the shadow  is behind the box.  The shadow extends from the bottom of the subject matter where it touches the plane.
12. Finally, remove the guidelines and fill in the shadow with a dark tone, relative to the brightness and quality of light you would like to represent.  I kept the angle of light lines active, so you can see how the corners of the shadow will meet them.  

Thank you for reading this week’s posting.  Next week I will show you how to cast shadows in perspective from a light source which is Before the Viewer.

Best wishes.