Quick Solvers 2: Amphitheater Arc Solver

In Quick Solvers 1, I showed three tabs from my solver spreadsheet that helps with common design and tuning tasks. Now it’s time to add another tab to that spreadsheet to help with audience geometry. The basic geometry that we use to model audience listening planes for prediction often tells us everything we need to know despite sometimes being quite simplified from reality. With a few shows of a laser disto, the large flat floors and raked seating inclines are easy to build in most prediction platforms. It’s the curved surfaces that prove perennially difficult. As discussed in Dimensional Focus, sometimes we can dodge this bullet by leaving them out (after all this is how the early days of line array prediction were done - all 2D landing strips).

Sometimes, though, we can’t safely omit all the curved surfaces from our prediction. What if the entire audience plane is curved? This type of “swept span” architecture is quite common in theaters, amphitheaters and the indoor-outdoor echo chambers we affectionally call ‘sheds,’ in which the audience geometry is effectively an inclined plane pivoted or extruded around a point, as if it were turned on a lathe.

For example, we could model the venue below as a simple on-axis landing strip geometry to reasonably determine our trim heights and splays, but we’d need the full 3D geometry to assess center gap, horizontal horn patterns and so forth.

Figure 1 - Saratoga Performing Arts Center, Saratoga Springs NY

You Say You Want A Revolution?

Getting this geometry created can be far more time consuming than the typical flat and raked rectangle or triangle sections we work with in clubs and arenas. Since these types of shapes are typically modeled by finding the center point of the (mathematical) circle that serves as the center point for the geometry rotation, this task is often approached using a three-point circle calculator. However, finding the X/Y coordinates for three points in the seating area that can represent the perimeter of the circle is often not the most straightforward task, especially in venues without straight lines or right angles to reference off. In some cases this center spot can be found visually by standing on stage at the visual intersection of the seating aisles, but that relies upon a few assumptions, and sometimes the circle center point lies in the artist’s green room behind the stage, which is distinctly unhelpful. Even if we find the center of the circle without much issue, finding the angle span of the seating area poses a puzzle.

In SoundVision,

this type of geometry is referred to as a Revolution. Simply entire the depth and height to the front and rear edges of the shape - but we still need a “Described angle,” or how many degrees of revolution to apply to the shape. As it turns out, this is a platform-agnostic question. If we are working in other prediction software - EASE Focus, MAPP 3D, ArrayCalc - we still need to know the center of revolution and the angle of span to create these shapes, and they can be tricky to measure quickly in the field with just a laser.

Figure 2 - An example "Revolution" geometry object in SoundVision.

Another one of those math formulas you learned about in high school and then promptly forgot rushes to the rescue. Turns out we can get the values we need - center point for the revolution and the angle of span - with just a few up-down and left-right laser shots, which are quick and easy, and then entering that data into a solver I wrote that uses the Slope Intercept Form of the basic line equation to calculate everything we need to know to draw the shape.

The specific input parameters we need depend on which prediction platform we’re using because they all use slightly different methods for creating geometry, however the basic concept is very simple: We will walk down the center line of the venue, and shoot two laser shots straight out to the edge of the seating area. For each shot, that measured distance will form a Y coordinate, and the distance from the stage will form the X coordinate. I call the first of these two shots the Minor Ruler, and the second (longer) one the Major Ruler. Figure 3 below shows an example of these two laser shots. The exact depth at which you choose to take these measurements, somewhat counterintuitively, doesn’t matter, as long as they’re perfectly perpendicular to the venue’s center line (left-right) and are accurate to the edge of the geometry we wish to draw.

Figure 3 - From the center line of the venue, measure two laser shots to the edge of the geometry. For each, this distance is the Y value, and the distance from the stage is the X value. 

Here’s a real-world example thanks to Google Earth Pro, an aerial view of a small outdoor amphitheater seating geometry. (Figure 4)

Figure 4 - an overhead view showing an example of the laser shots and dimensions required for the calculations

Based on these measurements, our Minor Rule coordinates would be (21, 84) and the Major Rule coordinates would be (66, 127). If the seating area is raked (as it probably is) you will also want to take notes of the inclination as well. Some measurement tools will give you the angle of rake, which you can use with a basic right triangle calculator to get the z height coordinate, and some measurement tools will solve for vertical height directly and do the work for you.

Entering these two pairs of coordinates into the Major Rule and Minor Rule X/Y fields in the solver spreadsheet will calculate the slope of the line and its x intercept, from which it can calculate the span angle.

If you’re using SoundVision, simply create a new Revolution, enter the depth and height of the front and rear edges of the geometry as measured from (0,0), and use the Span Angle generated by the solver for the Described Angle and you’re off to the races.

In MAPP 3D,

the easiest way to draw these geometries is to start by drawing the on-axis center line vertical elevation profile (“landing strip”) as you would for a simple 2D prediction, and then calculate the Angular Extrusion using the Angle and X Center values from the solver.

For example, we can insert a Free Draw Geometry to represent the basic profile of this seating rake: front of the seating area 8 feet from the stage, elevation of 3 feet (Point 1: 8, 0, 3) and the rear of the seating area at 260 feet from the stage, elevation of 12 feet (Point 2: 260, 0, 12). it doesn’t look like much (Figure 5, left) but inputting the values for the Major and Minor rulers gives us extrude parameters of 43.7 degrees and -67 feet for the X center. Let’s do an angular extrusion (Figure 5, middle), then make it a prediction plane and mirror it over XZ to get the complete shape (right).

Figure 5

This can easily be expanded to more complex, multi-tiered audience geometry. For example, we can quickly draw the audience geometry for SPAC (shown in Figure 1) by creating a Free Draw Geometry using the following on-axis vertical profile points: Point 1: (22, 0, 3) Point 2: (87, 0, 9) Point 3: (138, 0, 16) Point 4: (136, 0, 29) Point 5: (91, 0, 29) Point 6: (145, 0, 53). If you were standing at the downstage (0,0) edge in the venue with your laser disto, these values would be quick and easy to gather by shooting at the audience geometry. (Figure 6)

Figure 6 - Seating profile dimension values for SPAC

Inserting this Free Draw Geometry creates the on-axis profile object that we can then extrude using our solver (Figure 7).

Figure 7 - the seating profile geometry

When I measured this venue, I gathered Minor ruler values of (40.1, 51.8) and Major ruler values of (90, 82.4) so let’s punch those in, and the calculator gives us a 31.5 degree angle and an X center point of -44.4. Perform an angular extrusion on the object with these parameters (Figure 7, left) and then add the appropriate prediction planes and mirror the shape to create the full geometry (Figure 8, right).

Figure 8 - Left: Extruded geometry. Right: Prediction planes mirrored

In ArrayCalc,

instead of extruding a 2D shape, we will create the geometry all at once like we did in SoundVision, and we will need the same two additional measurements: the distance from the (0,0) point (downstage center) to the front of the audience area (first row) and to the rear of the audience area (last row). Those all go into the appropriate fields in the solver spreadsheet, then we can create a new Arc Segment in ArrayCalc and fill in the appropriate parameters: The P0 (front, center) point, Start Span, Span, and the Inner and Outer radius values are all generated.

Note that the Major and Minor axis values will be the same if the shape is an annular or circular segment, as in all the examples from this article. If we’re dealing with oval-shaped geometry instead, those values would be different from each other (another solver, for another day, maybe). And don’t forget your rear Z height! Then, again, mirror the shape over the x axis and you’re done. (Figure 9)

Figure 9 - the same geometry in ArrayCalc

If you’re still following along, you can see that a few quick laser shots and the basic y=mx+b line formula give you everything you need to quickly create curved / annular raked geometry sections without much fuss.

Quick Solver Spreadsheet (Published Google Sheet | Download as Excel)