Closed-form sag solutions for Cartesian oval surfaces

Abstract

This paper presents a closed-form solution to the sag of the Cartesian oval and an alternate iterative method for obtaining the sag. The emphasis is in providing a methodology for determining the sag and derivatives of a Cartesian surface for optical design, ray-tracing purposes. We verify our results by comparison of our solutions and by real ray tracing.

Appendices

Appendix A

This appendix provides details on programming the Cartesian closed-form sag solutions:

• Step 1: Define π as “3.14159265358979323846”

• Step 2: For convention an object to the left of the surface has a negative object distance. The image distance is positive if the image is to the right of the surface.

• Step 3: Define the values s₁, s₂, n₁ and n₂.

• Step 4: Determine the coefficients A, B, C, D, and E.

• Step 5: Determine the coefficients α, β, and γ.

• Step 6: Determine the coefficients P and Q.

• Step 7: Determine the following parameters:

  $$ \delta = \sqrt {{ - \frac{c}{{3a}}}} = \sqrt {{ - \frac{P}{{3a}}}} . $$

  (A1)
  $$ h = 2a{\delta^3} $$

  (A2)
  $$ {\theta_1} = \frac{1}{3}{\cos^{{ - 1}}}\left( { - \frac{d}{h}} \right) = \frac{1}{3}{\cos^{{ - 1}}}\left( { - \frac{Q}{h}} \right) $$

  (A3)
  $$ {{\rm v}_1} = 2\delta \cos \left( {\frac{{2\pi }}{3} - {\theta_1}} \right),\;{{\rm v}_2} = 2\delta \cos {\theta_1},\;{{\rm v}_3} = 2\delta \cos \left( {\frac{{2\pi }}{3} + {\theta_1}} \right) $$

  (A4)
  $$ y = {\rm v} - \left( {5/6} \right)\alpha $$

  (A5)

• Step 8: Use Table 1 to calculate desired solution z _i = (α, β, y, A, B).

Appendix B

The factors kk and ww are for controlling the algebraic signs while the conjugate distances are positive or negative. When the conjugate distance s₁ is negative and the other conjugate distance s₂ is positive, the factors kk and ww are given a value +1. On the other hand, these factors become −1 when the sign of s₁ and s₂ reverse. To avoid handling large numbers that leads to errors as they are subtracted four cases for calculating the OPL are considered.

• Case 1: s₁ ≤ 10⁸

  $$ OPL = {{\text{n}}_1}\sqrt {{{{\left( { - {s_1} + z} \right)}^2} + {r^2}}} + kk \cdot {{\text{n}}_1}{s_1}. $$

  (B1)

• Case 2: s₁ > 10⁸

  $$ OPL = {{\text{n}}_1}\left( {z - \left( {{r^2} + {z^2}} \right)/2{{\text{s}}_1}} \right). $$

  (B2)

• Case 3: s₂ ≤ 10⁸

  $$ OPL = {{\text{n}}_2}\sqrt {{{{\left( {{s_2} - z} \right)}^2} + {r^2}}} - ww \cdot {{\text{n}}_2}{s_2}. $$

  (B3)

• Case 4: s₂ > 10⁸

  $$ OPL = {{\text{n}}_2}\left( { - z + ({r^2} + {z^2})/2{{\text{s}}_2}} \right). $$

  (B4)

where r in the four cases represents $\sqrt {{{x^2} + {y^2}}}$ and x and y are the coordinates of intersection of a given ray.

Appendix C

The slope m _x and m _y of the tangent line at a given point can be calculated once the sag at that point is known as follows,

• $$ {\text{numerator}} = \frac{{ - {n_1}{s_1}/{n_2}{s_2}}}{{s_1^2\sqrt {{{{\left( { - 1 + z/{{\text{s}}_1}} \right)}^2} + {r^2}/s_1^2}} }} + \frac{1}{{s_2^2\sqrt {{{{\left( {1 - z/{{\text{s}}_2}} \right)}^2} + {r^2}/s_2^2}} }}. $$

  (C1)
• $$ {\text{denominator}} = \frac{{\left( { - {n_1}{s_1}/{n_2}{s_2}} \right)\left( { - 1 + z/{{\text{s}}_1}} \right)}}{{{s_1}\sqrt {{{{\left( { - 1 + z/{{\text{s}}_1}} \right)}^2} + {r^2}/{\text{s}}_1^2}} }} - \frac{{\left( {1 - z/{{\text{s}}_2}} \right)}}{{{s_2}\sqrt {{{{\left( {1 - z/{{\text{s}}_2}} \right)}^2} + {r^2}/{\text{s}}_2^2}} }}. $$

  (C2)
• $$ {{\text{m}}_{\text{x}}}{ = } - x \cdot \frac{\text{numerator}}{{\text{denominator}}}. $$

  (C3)
• $$ {{\text{m}}_{\text{y}}}{ = } - y \cdot \frac{\text{numerator}}{{\text{denominator}}}. $$

  (C4)