6
SLOPING PLANES AND SURFACES
Rectilinear planes that are parallel to the ground plane will have vanishing points that fall on the horizon line.
If one axis of a receding plane is not parallel to the ground plane, its vanishing point will not fall on the horizon line. Instead, it will fall on a line perpendicular to the horizon line that runs through the original vanishing point. This line is called the vertical vanishing line (VVL). Vanishing points that fall on this line are called vertical vanishing points (VVP).
The steeper the angle of a plane's ascent or descent from the ground plane, the farther up or down on the vertical vanishing line the points will fall.
It is important to realize that the vertical vanishing line operates just like the horizon line, except that it is perpendicular to the horizon.
Turn the image on end and note that the image becomes a three-point-perspective setup.
Note also how the sloping planes diminish toward vanishing points below the horizon line after they have passed an angle 90 degrees to the ground plane.
DRAWING SLOPES OFF RECTANGLES
It is not always convenient or necessary to find the vanishing points for a sloped plane or angle.
If the base and the height of an angle are known, the angle can be drawn by connecting the two extremes with a diagonal line.
If the base is drawn in perspective, the sloping plane will automatically converge toward its vertical vanishing point. Thus, it is possible to plot complex angles and slopes by determining their base length and height, as illustrated in the examples that follow.
Examples
Sketching Slopes
Intersecting Sloping Planes
When two planes that are perpendicular to the ground plane intersect with each other (A), they form a corner that is also perpendicular to the ground plane (B).
When one of the intersecting planes is at an angle to the ground plane (A), that angle will be described on the perpendicular plane where the two planes intersect (B).
If both of the intersecting planes are at an angle to the ground plane (A), the corner at which they intersect will be a compromise between the two angles (B).
If the tops of intersecting planes are the same height off the ground plane, the angle of the intersecting corner can be found by drawing a line between the points where the edges of the two planes meet.
However, if the intersecting planes are of different heights, it is necessary to find the point where the smaller plane enters the larger one before the position and angle of the corner can be drawn. See the following directions.
To find the point on the larger plane where the smaller one intersects it, mark the height of the smaller plane on the larger plane. This point marks the top of the intersecting corner.
If the vertical vanishing point for the larger slope is known, a line can be extended from the top of the smaller plane perpendicular to the ground, across toward its own vanishing point, and then up to the larger slope's vertical vanishing point.
Sloped Planes
Sketching Intersecting Slopes
DRAWING A MEASURED ANGLE IN PERSPECTIVE
When viewed in perspective, a plane that is at an angle to the ground plane will have proportions different from its horizontal counterpart because of the change in distance between the observer and the object.
The geometric consistency of linear perspective makes it possible to determine the following slope characteristics:
- The angle of the slope in degrees
- The length and proportions of the receding slope, to scale
In order to accomplish this, it is necessary to find and use measuring points (see Chapter 4, “Constructing a Two-Point-Perspective Grid”). Use the following procedure.
Begin by finding the measuring point for the axis on which the angle will ascend and descend.
Once you have transferred the vanishing points and measuring points to the horizon line, lay down a ground line and connect the base lines of the angle to their respective vanishing points.
The measuring point provides a side view (elevation) of the angle, so the angle can be scaled and measured with a protractor. Here, the angle is 25 degrees.
Note that all angled lines that strike the 25-degree vertical vanishing point are 25 degrees off the ground plane, no matter where they fall.
Since any angle can become the diagonal of a rectangle, the vanishing point of an angle can serve as a guide for multiplying and dividing rectangles of given proportions.
DRAWING A MEASURED SLOPED PLANE IN PERSPECTIVE
To draw a measured sloped plane in perspective, follow these steps:
Draw a rectangle to scale by using a scaled ground line and appropriate measuring points, as shown in Chapter 4, “Constructing Perspective Grid Systems.” Here, the rectangle is 10 feet by 10 feet; the desired angle is 25 degrees.
The angle of the slant will be 25 degrees above the horizon, so a vertical measuring line should be drawn perpendicular to the ground line where the two base lines meet. Mark this line with the same scale increments as the ground line.
Establish the sloping plane by connecting the right-hand corners of the rectangle to the 25-degree vertical vanishing point.
In order to mark this slope to scale, you must find a measuring point on the vertical vanishing line. Draw a line from this point to the scaled vertical measuring line (VML) to mark the limit of the sloping plane.
Note that the vertical measuring point (VMP) is found by swinging an arc down from the measuring point on the horizon line, which has its axis at the 25-degree vanishing point.
Turn this view 90 degrees and you will see that the vertical vanishing line becomes a horizon and the vertical measuring line becomes a ground line.
If you have a lot of sketches to do using the same slope, it is useful to estimate the position of the vertical vanishing point off the paper and swing an arc around it. Then sketch in some radii as guides for your slope lines as shown in the illustration “Sketching Estimated Slopes.”
Sketching Estimated Slopes
