I have been exploring how to draw buildings in perspective for a comic I would like to make called The Adventures of Stilton. The buildings can be drawn as cubes, rotated to different angles. A cube is a shape where identical square faces join to neighbouring faces with an internal angle of 90 degrees. However the issue is that the cube will appear to have differently proportioned faces and a range of angles between those faces when it is seen in perspective and those angles and proportions will appear to change yet again as the cube is rotated. So it is clear that some strategies are needed to build up information to represent a cube in perspective and rotated in perspective.
In this post I thought I would describe one of the methods I have been using to rotate a cube in perspective. The method begins with what I call a framing square which is just a square drawn in one point perspective. The framing square is double the width of the cube I want. Within the framing square I can construct identical cubes in eight different positions. This video shows the eight different cube positions in the framing square. One in each of the four corners of the framing square by dividing the square horizontally and vertically in perspective, and one in each of the four divisions when the framing square is divided diagonally. All eight cubes have an edge fixed at the centre of the framing square, I call it the axis edge.
As a brief outline to how the cubes are built up they can be considered to be made of following pieces: the axis edge, that is the same for all the cubes, then the face of the axis edge, for the four cubes that fit exactly within the framing square the cubes are constructed from the corresponding edges of the framing square and for the cubes on the diagonal divisions of the framing square, the face with the axis edge traces a circle that fits perfectly within the framing square and then the remaining faces of these four cubes can be constructed from a two point perspective that comes from drawing of the framing square. That outline will be so much clearer as I go through the more detailed steps of constructing these cubes.
To begin I draw the framing square in perspective. I start with a blank sheet of paper and I draw a cross made of a horizontal and vertical line as I described in my post for March this year. There seems to be a convention of calling the horizontal line the horizon line and the vertical line the line of sight and the point they intersect the view point, so those are the terms I'll use. Then I choose a point to fan out two lines at 45 degrees from the line of sight so that the 45 degree lines go on to cross the horizon line. The intersection points of the 45 degree lines and the horizon line I shall call the 45 degree points, they become useful for the next part.
The front of the square can be drawn anywhere parallel to the horizon line but for this drawing I place the horizontal line equally divided by the line of sight. I also like to make the horizontal line a length that I can easily divide into ten equal parts, this will be useful later when I draw a circle within the square. Then I send two lines from either side of the front of the square to the view point, these are the sides of the square. The sides of the square are now useful to determine the depth or where the back of the cube will be. Drawing a line from the front corner of the square to the opposite 45 degree point on the horizon line will cross the newly drawn side of the square. The place this line crosses the square side is the back of the square, so I could just draw a line from this point that will be parallel with the horizon line and have completed the square. However I like to repeat the process on the opposite front corner of the framing square so that I have two points, one on either side of the square's sides that I can join to form the back of the square. The benefit of this extra step is that I get two diagonal lines from the front corners of the square to the opposite back corners and the point these lines cross is the centre of the framing square in perspective, which is a very useful thing.
At this point I could draw the four cubes in the corners of the framing square, beginning with the axis edge that all the cubes share. However I prefer to leave the construction of the axis edge until after I have drawn the circle within the framing square as drawing the circle, which in perspective appears as an ellipse, finds all the reference lines I need for the axis edge along with it.
I draw a circle in the framing square by dividing the square up into a grid. To do this I follow a method described in this Dan Beardshaw video about drawing a circle in perspective. A grid can be formed in the square by drawing lines from ten equally spaced points on the front edge of the square. The vertical lines of the grid simply extend from the ten equal unit points to the view point. The horizontal lines are drawn with a method similar to the way the back of the framing square was identified: from each of the ten unit points on the front of the square I draw a line to one of the 45 degree points. It can be either of the 45 degree points but all ten lines need to go to the same point. Then the intersection from each of these lines and side of the framing square nearest to the 45 degree point chosen marks the position of the horizontal lines of the grid. There is a nice check here that the fifth of these lines pass through the centre of the framing square defined also by the diagonal lines crossing from the front to the back of the framing square.
With this ten-by-ten square grid in perspective I have not only the middle of the square marked out horizontally and vertically by the fifth of the ten horizontal lines, which are all points where the ellipse will touch, but I also have some guide points for the ellipse between these points. The ellipse will pass diagonally through the grid squares that are one column in and one row up from each corner of the square. I like to lightly shade these squares as they are useful points of reference that I will call the shaded squares. With all this information and some care I draw an ellipse.
Along with the horizon line, the view point and the 45 degree points, I now have the following all in perspective: the framing square, the circle within and two ways to identify the centre of the framing square, through the crossing of the diagonal lines and the crossing of the fifth of the ten grid lines. This is all I need to construct all eight cubes.
To construct the four cubes that fit exactly within in the framing square when it is divided into four equal squares and will be drawn in the same one point perspective as the framing square, I need the framing square and the fifth of the ten lines from the grid that divides the framing square into four equal spaces in perspective and the view point. I shall now refer to the squares formed by dividing the framing square by the fifth grid lines as the cube bases. These cube bases have the quality that their width is equal to the cube's height from this perspective. So I draw two vertical lines from each of the cube base's corners that are the same height as the cube base's width. The new sides of the cube then are connected with a horizontal line that forms the cube's top front edge. Extending lines from both of the top front corners of the cube to the view point gives the top sides of the cube. I use a transparent ruler to get the vertical lines to be perpendicular to the horizontal lines of the framing square. In constructing these four cubes I have also drawn the axis edge that is needed for drawing the next four cubes.
To draw the cubes aligned with the diagonal divisions of the framing square, that is within the lines that defined the depth of the framing square and extend from the front corners of the framing square to the 45 degree points on the horizon line, I need the framing square, the circle that appears as an ellipse within the framing square, the 45 degree points on the horizon line and the axis edge.
The cube face with the axis edge already has two sides in place, the axis edge itself and the diagonal line up to the point of ellipse, to complete this cube face I draw a line from the top of the axis edge to the 45 degree point that is closest to the ellipse edge we are drawing to, this forms the top edge of the cube face. The face is completed with a vertical line up to the top edge from the ellipse. The 45 degree points on the horizon line now function as the vanishing points to draw the remaining cube faces in two point perspective.
Now I have a drawing of eight identical cubes each at different angles and in perspective. Drawing comics is where you imagine something and write and draw it into being, so you meet your imagination with a reality of your own creation. It is a full circle. Learning some perspective with a bit of care gets me just a little bit closer to that full circle.