Chapter 7 of 8
Chapter 7: Spherometers and Diopter Gauges
7.1 Introduction
Not all engineering surfaces are flat or cylindrical. In optics, precision manufacturing, and advanced inspection, engineers must measure curved surfaces accurately. Spherometers and diopter gauges are specialized instruments designed for this purpose. They are used to determine the radius of curvature and optical power of spherical surfaces such as lenses.
This chapter introduces spherometers and diopter (Geneva) gauges, explains how they work, and shows how measurements are converted into meaningful engineering quantities. Students will connect physical measurements to geometry, algebra, and Excel-based analysis.
7.2 Radius of Curvature and Optical Power
Radius of Curvature (R)
The radius of curvature describes how strongly a surface curves. A small radius indicates a tightly curved surface, while a large radius indicates a gently curved surface.
- Convex surface → positive radius
- Concave surface → negative radius
Optical Power (Diopters)
Optical power describes how strongly a lens bends light and is measured in diopters (D).
- Convex lenses → positive power
- Concave lenses → negative power
For laboratory use with typical optical materials, the relationship between power and radius is approximated by:
R (mm) = 350 / P
P (diopters) = 350 / R
This constant is an instructional approximation — see Note 6 in Notes & Clarifications.
7.3 Spherometers
Spherometers are used to measure the radius of curvature of spherical surfaces by measuring the sagittal height (h) of a surface relative to a known base diameter (d).
A spherometer consists of:
- A calibrated ring base (three-point support)
- A central dial gauge or depth gauge
- Interchangeable bases of different diameters
Larger bases generally produce more accurate results.
7.4 Sagittal Height (h)
Sagittal height is the vertical distance between the center of the surface and the plane defined by the spherometer base.
- Convex surface → positive h
- Concave surface → negative h
Correct zeroing on an optically flat surface is critical before measuring sagittal height.
7.5 Spherometer Radius Equation
The radius of curvature is calculated using:
R = d² / (8h) + h / 2
Where:
- R = radius of curvature
- d = base diameter
- h = sagittal height
This equation is derived from circle geometry and assumes a spherical surface.
7.6 Selecting the Correct Spherometer Base
Proper base selection is essential:
- Use the largest base that fits entirely on the lens surface
- For convex surfaces, use the inner diameter (ID) of the ring
- For concave surfaces, use the outer diameter (OD) of the ring
Different bases may be required for opposite sides of the same lens.
7.7 Diopter (Geneva) Gauges
A diopter gauge directly measures optical power rather than sagittal height.
Key features:
- Dial gauge mounted in a ring base
- Black scale for convex (+) lenses
- Red scale for concave (−) lenses
The gauge reading indicates diopter power, which can be converted into radius.
7.8 Reading a Diopter Gauge
General rules:
- Clockwise vs. counterclockwise motion depends on zero orientation
- Sign (+ or −) determines convex or concave surface
- Always identify zero before reading
Radius conversion:
R (mm) = 350 / P
7.9 Meniscus Lenses
Meniscus lenses have:
- One convex surface
- One concave surface
The total power of a meniscus lens is the algebraic sum of the two surface powers:
Ptotal = Pconvex + Pconcave
A positive total power produces magnification.
Try It: Spherometer Radius Calculator
Enter the base diameter and sagittal height in millimeters. Use negative h for concave surfaces.
Try It: Diopter ↔ Radius Converter
Uses the course constant R (mm) = 350 / P.
7.10 Excel Integration
Excel is used to:
- Enter spherometer equations
- Automate radius calculations
- Compare results from different bases
- Calculate percent error
- Create scatter plots showing base diameter vs. error
This reinforces engineering data analysis and repeatability.
7.11 Chapter Summary — Study Points
- Spherometers measure sagittal height
- Radius of curvature depends on base diameter and h
- Larger bases improve accuracy
- Diopter gauges measure optical power directly
- Convex is positive, concave is negative
- Meniscus lenses combine powers
- Excel is required for analysis
Chapter 7 Exercises
Each exercise includes a worked example followed by a student practice problem.
Exercise 1: Identifying Surface Type
Example (Solved)
Diopter reading = +5.00 D
Surface type: Convex
Student Practice
Diopter reading = −3.50 D. Identify the surface type.
Show answer
Concave surface — negative power means concave.
Exercise 2: Radius from Diopter Reading
Example (Solved)
P = +10.0 D
R = 350 / 10.0 = 35.0 mm
Student Practice
P = −7.0 D. Calculate the radius of curvature.
Show answer
R = 350 / (−7.0) = −50.0 mm (concave).
Exercise 3: Sagittal Height Method
Example (Solved)
d = 50 mm
h = +0.40 mm
R = (50² / (8 × 0.40)) + (0.40 / 2)
R = (2500 / 3.2) + 0.20 = 781.25 + 0.20
R ≈ 781.5 mm
(Corrected in this edition: the original printing gave 78.3 mm, a factor-of-ten error.)
Student Practice
d = 60 mm
h = −0.30 mm
Calculate the radius of curvature.
Show answer
R = (60² / (8 × (−0.30))) + (−0.30 / 2)
R = (3600 / −2.4) − 0.15 = −1500 − 0.15
R ≈ −1500.2 mm (concave)
(Corrected in this edition: previously printed as −150.2 mm.)
Exercise 4: Meniscus Lens Power
Example (Solved)
Convex side = +6.00 D
Concave side = −2.00 D
Total power = +4.00 D
Student Practice
Convex side = +8.50 D
Concave side = −3.25 D
Determine the total power.
Show answer
Ptotal = +8.50 + (−3.25) = +5.25 D
Exercise 5: Base Selection
Example (Solved)
Lens diameter = 120 mm
Select the largest ring base smaller than the lens diameter.
Student Practice
Lens diameter = 95 mm. Explain which ring base should be selected and why.
Show answer
Use the largest ring base with a diameter less than 95 mm so the entire ring sits on the lens surface. Larger bases produce a larger sagittal height for the same curvature, which reduces the percent error caused by small errors in reading h.
Chapter 7 Resources
- MET-135-Lab-07 — Spherometers and Diopter Gauge
- Spherometer and Geneva Gauge equations (course handout)
- Lens measurement examples (course materials)