Concentrator Geometry

Introduction

Although this is written as a practical guide for parabolic trough designers, it seemed a good idea to review some plane and analytic geometry basics and vocabulary.

Plane Geometry

A parabola is defined as the locus of points equidistant from a fixed point (called the focus) and a line (called the directrix).

The line passing through the focus and perpendicular to the directrix is called the parabola's axis of symmetry.

The line passing through the focus and parallel to the directrix is called the latus rectum.

The vertex of a parabola is the point on the axis of symmetry midway between the focus and the directrix.

The focal length of a parabola is the distance between its focus and vertex.

Constructing a Parabola with Compass and Straightedge: Red = compass, Blue = straightedge, Black = parabolic curve 1. Draw the directrix (bottom blue line) 2. Draw vertical line from directrix to focus (left blue line) 3. Mark midpoint of vertical line (vertex) 4. Set compass greater than or equal to focal length 5. Mark this distance from directrix on vertical blue line 6. Scribe line parallel to directrix through this point 7. Scribe arc centered at focus to intersect this line 8. Repeat from step 4 with new length until you have enough points 9. Draw lines to connect the intersections (black curve)

Analytic Geometry

An upward facing parabolic curve is defined by the equation y = (x − h)² / 4a + k, where (h, k) are the coordinates of the vertex and a is the focal length; and the slope of a line tangent to this parabola at any point will be
dy/dx = (x − h) / 2a

When the vertex is located at the origin, that equation can be simplified to y = x² / 4a; and the slope of a line tangent to this parabola at any point will be dy/dx = x / 2a.

Example 1 (Chord + Absorber Displacement)

The starting parameters are a horizontal chord (the aperture width of the mirror) of length L, the radius of the target tube, and the distance between the target tube and a glass cover.

Step 1: Calculate c = (radius of the target tube) + (clearance distance), and locate the focus c units below the midpoint of the chord. Step 2: Determine the distance d between the focus and an end point of the chord. Locate the directrix this distance below the chord. Notice that d² = c² + (L / 2)² [Pythagorus] and so d = sqrt(c² + (L / 2)²).

Step 3: Locate the vertex of the parabola halfway between the focus and the directrix. The distance between the vertex and the focus is a, the focal length of the parabola. Now we can calculate a = (d − c) / 2. Step 4: If the vertex is located at the origin, then the equation is y = x² / 4a. The plotted parabolic curve will pass through both endpoints of the chord and the vertex. Plot the curve from −L/2 to +L/2 using the focal length calculated in step 3 above. Step 5: All that remains is to add some structural elements to support the mirror, and to show the target tube. Example 2 (Chord + Point on Parabola)

The starting parameters are a chord (aperture width) length of 96 inches and a point on the parabola at x = 48, y = vertex + 12. Let's locate the vertex at the origin to simplify the equation to y = x² / 4a, which will locate the given point at (48,12).

If we transpose the terms of the equation, we can produce a = x² / 4y; and if we substitute 48 for x and 12 for y, we can calculate the value a = (48)² / (4 * 12) = 48.

When we substitute this value for a into the general equation for a parabola with its vertex at the origin, we find that the equation for the parabola we want is y = x² / 192. 