Choosing a stable projection range
Scanning a concave 20 cm mirror from 25 to 120 cm shows rapid variation near the lower edge and progressively steadier screen positions farther beyond 2f.
Physics and geometric optics
Mirror Graph Calculator
Graph signed image distance and magnification against real-object distance for a concave or convex spherical mirror.
Object-distance response curve
See where mirror response changes gradually, sharply, or without bound
This page samples the signed spherical-mirror equation across a chosen real-object range. It explicitly breaks a concave curve at the focal singularity instead of drawing a false line through infinity.
Reference image distance-
Reference magnification-
Reference image height-
Sampled finite span-
Current model evidence
Mirror response-curve ledger
Link the highlighted reference result to a domain-aware scan that never bridges a focal asymptote.
| Curve basis | Lower / current value | Upper / focal value | Derived value | Scope / unit |
|---|
DETAILED CALCULATION PROCESS
Formula, units, default substitution, and reconciliation
1. Governing relation
di(do) = f do/(do - f); m(do) = -di/do = -f/(do - f)
Assign the signed focal length, sample only positive real-object distances, calculate signed di and m at each sample, insert a gap around a concave focal singularity, and evaluate the reference with the same direct equations.
2. Symbols and default basis
| Symbol | Meaning | Unit | Default basis |
|---|---|---|---|
| do | Scanned real-object distance | cm | 25 to 120 cm |
| f | Signed focal length | cm | +20 cm for concave |
| di(do) | Signed image-distance response | cm | Curve output |
| m(do) | Signed magnification response | 1 | Curve output |
| do,ref | Highlighted object distance | cm | 60 cm |
| ho | Reference object height | cm | 5 cm |
3. Unit and sign normalization
- Range, focal magnitude, reference distance, and height all use centimetres.
- Mirror type supplies the focal sign; the focal-magnitude field remains positive.
- Null graph samples are intentional domain gaps, not zeros or missing calculations.
4. Current numerical substitution
5. Independent reconciliation
HOW TO USE THIS CALCULATOR
Five steps from measured inputs to a defensible result
- Choose the mirror type and enter a positive focal magnitude.
- Set a positive object-distance interval with the minimum below the maximum.
- Place the reference distance inside that interval and away from a concave focal point.
- Use the image-distance panel for bench placement and the magnification panel for size and orientation regime.
- Inspect the asymptote and exact reference ledger before interpolating any design decision from the curve.
MIRROR PHYSICS FOUNDATIONS
Concepts that control this specific model
- The response is hyperbolic
- Both di and m depend on 1/(do-f), so equal object steps do not produce equal image changes.
- A concave focal point divides regimes
- Below f the image is virtual and upright; above f it is real and inverted.
- Convex response has no positive-object singularity
- With f < 0 and do > 0, the denominator do-f stays positive.
- Signed axes retain physical meaning
- Negative image distance and positive magnification identify virtual upright images.
- A plotted sample is not a new formula
- The highlighted point and curve use the same mirror equation; sampling only reveals its behavior over a domain.
DEEP ANALYSIS 1
Never connect across infinity
A straight segment between large positive and negative di values would imply a finite path through the focal point. The visual uses a gap and asymptote marker.
DEEP ANALYSIS 2
Graph scaling must not rewrite results
Extreme near-focus samples can dominate the canvas. Display clipping improves readability, while the exact reference value and ledger retain the unrounded model.
DEEP ANALYSIS 3
Sensitivity is encoded by slope
Near f a small do measurement error can cause a large di change. A steep curve is therefore a warning to quantify tolerances before placing a screen.
RESULT INTERPRETATION
What the current output does and does not decide
The reference cards are exact direct evaluations at do,ref; use the curve to understand nearby behavior rather than reading a low-resolution pixel value.
The finite sampled span summarizes the selected numerical samples and is not a physical limit on image distance near a singularity.
REAL USE CASES
Two decisions with different boundary conditions
Surveying a convex viewing mirror
A convex scan remains virtual and upright for every positive object distance, approaching a smaller-magnitude negative image distance as the object moves farther away.
EVIDENCE AND DATA QUALITY
What to retain with the exported result
Retain the selected domain, sample count, focal sign convention, exact reference point, curve-gap rule, object-distance measurement method, and aperture conditions.
LIMITS AND EXCLUSIONS
Where this physical model stops
- Uses 181 explanatory samples rather than adaptive optical ray tracing.
- Excludes the concave focal singularity from finite graph points.
- Applies only to paraxial spherical mirrors and one real object.
- Canvas display clipping does not change exported exact values.
- Does not model aperture, aberrations, reflectance, diffraction, or uncertainty bands.
TERMS USED HERE
Six terms that keep the calculation unambiguous
- Response curve
- Output evaluated across a controlled input domain.
- Asymptote
- Line approached without a finite value at the focal singularity.
- Domain gap
- Intentional break where the finite equation result is undefined.
- Reference point
- Exact current object distance marked on each curve.
- Signed image distance
- Positive for real convergence and negative for a virtual image.
- Sensitivity
- Rate at which an output changes for a small input change.
RELIABLE SOURCES
References supporting the equation and units
FREQUENTLY ASKED QUESTIONS
Questions specific to this calculation
Why is there a break in a concave graph?
At do = f the paraxial image is at infinity, so connecting the two finite branches would be physically false.
Why must the reference stay inside the range?
The marker and exact cards are intended to explain the displayed domain; an external reference would not be visible.
Can the range start below the focal length?
Yes. That includes the virtual-image branch, provided the graph treats the focal point as a gap.
Why does the convex curve not cross infinity?
For f < 0 and do > 0, do-f cannot be zero.
Is sampled finite span the maximum possible di?
No. It is only the range among finite samples and grows without bound as a concave sample approaches f.
Can I infer blur from the graph?
No. Blur requires aperture and aberration information that the paraxial equation does not contain.
IMPORTANT BOUNDARY
Use the result as analysis, not certification
This sampled paraxial response is an educational and first-order planning aid, not an aberration analysis, tolerance simulation, or optical-design certification.