Reflect a direction about the oriented normal
The law of reflection compares angles with a surface normal, not with the surface itself. A normal is perpendicular to the local reflecting surface. In the displayed plane, the incoming direction is u=(sinβ,−cosβ), while the outward normal is n=(sinα,cosα). The incoming arrow points toward the surface, so its outward-facing angle is measured using −u. A ray diagram that uses u directly as the outward incident arm can accidentally replace an angle by its supplement.
Learning outcome: Keep β=30° and compare α=−10°,0°,10°. Draw the normal before labeling angles and retain the incoming/outgoing vectors. Separately place a virtual plane-mirror object 3 cm in front and predict its image location and orientation.
Starting recap
Use normal-based angles, signed conjugate distances and explicit optical-path/reflection phases. Separate geometric direction, amplitude and energy; preserve focal and critical boundaries.
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Try
With β=30° and α=10°, incidence is 40° from the normal. The outgoing direction is 50° from upward vertical, with components (sin50°,cos50°). Its length remains one. Relative to the α=0 case the mirror normal turned 10° but the reflected ray turned 20°.
Open the experimentPredict
With a fixed incident ray, a 10° rotation of a flat mirror changes the reflected direction by:
Experiment
Keep β=30° and compare α=−10°,0°,10°. Draw the normal before labeling angles and retain the incoming/outgoing vectors. Separately place a virtual plane-mirror object 3 cm in front and predict its image location and orientation.
Reflect a direction about the oriented normal
Worked teaching model · synthetic values · Incident unit direction (sin beta,−cos beta), surface through origin, normal n=(sin alpha,cos alpha). Both incident and reflected paths lie in one plane. Front-face incidence throughout the grid. Equal spatial scales; no actual optical source or eye exposure.
Reset keeps your prediction, captures and practice history.
Incident and reflected ray directions with the tilted surface
Spatial axes use the same pixels per physical unit, preserving geometry. 1: incident toward the origin then reflected away (solid).2: surface tangent (dashed).3: oriented normal (dotted). Joined curves show the supplied model, not additional observations.
Equivalent numerical coordinates for Incident and reflected ray directions with the tilted surface
incident toward the origin then reflected away
| x (arbitrary length unit) | y (same length unit) |
|---|---|
| -0.5 | 0.866025 |
| 0 | 0 |
| 0.5 | 0.866025 |
At most 82 representative computed coordinates including endpoints; the plotted curve can use a finer grid. A table of a revealed generator is not independent measurement evidence.
surface tangent
| x (arbitrary length unit) | y (same length unit) |
|---|---|
| -1 | 0 |
| 1 | 0 |
At most 82 representative computed coordinates including endpoints; the plotted curve can use a finer grid. A table of a revealed generator is not independent measurement evidence.
oriented normal
| x (arbitrary length unit) | y (same length unit) |
|---|---|
| 0 | 0 |
| 0 | 1 |
At most 82 representative computed coordinates including endpoints; the plotted curve can use a finer grid. A table of a revealed generator is not independent measurement evidence.
| Quantity | Value | Unit |
|---|---|---|
| Incidence angle from normal | 30 | degrees |
| Reflected angle from upward vertical toward +x | 30 | degrees |
| Reflected unit direction x | 0.5 | ratio |
| Reflected unit direction y | 0.866025 | ratio |
| Reflected direction norm | 1 | ratio |
| Normal-component reversal residual | 0 | ratio |
Reflection reverses the normal direction component and preserves the tangent component.
For unit incoming u and unit normal n, outgoing v=u−2(u dot n)n. The incident ray travels toward the surface, so the angle to the outward normal uses −u. Here incidence is alpha+beta on the bounded nonnegative grid combination; the outgoing angle from upward vertical is 2alpha+beta. Rotating the normal by an increment rotates the reflected direction by twice that increment while the incident direction stays fixed. This is geometric reflection, not a diffraction or rough-surface model.
Predict before inspecting the values. Save contrasting records with assumptions and one alternative explanation. This virtual display does not certify practical measurement technique.
Reflection reverses the normal direction component and preserves the tangent component.
For unit incoming u and unit normal n, outgoing v=u−2(u dot n)n. The incident ray travels toward the surface, so the angle to the outward normal uses −u. Here incidence is alpha+beta on the bounded nonnegative grid combination; the outgoing angle from upward vertical is 2alpha+beta. Rotating the normal by an increment rotates the reflected direction by twice that increment while the incident direction stays fixed. This is geometric reflection, not a diffraction or rough-surface model.
Captures preserve recipes and replay their calculated results. They are synthetic model evidence, not physical measurements.
Understand
Apply
Public practice checks a specific claim. It does not establish independent mastery.
Which independent observation would reject the approximation, sign convention or calibration premise rather than merely refit it?
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Completion requires a saved prediction, at least one capture and a reflection. Presence checks do not grade your reasoning.
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