Shadow Geometry with Parallel Light and Dual Light Sources

Parallel Light and Similar Triangles

  • Sun is typically treated as a parallel light source because it is so distant; rays arriving at Earth are effectively parallel.
  • If you have a very tiny light source, the rays still come in at essentially the same angle, so the geometry reduces to similar triangles.
  • Key idea: similar triangles arise when light rays from a source interact with an opaque object and project onto a screen; the boundaries of shadows are governed by these similar triangles.
  • Notation introduced in the discussion:
    • Let l denote the line of a light ray (the path from the light source toward the screen).
    • Consider the object (opaque) and a screen placed somewhere in the light path.
    • Two light sources may be involved, producing separate shadow regions on the screen (e.g., red and blue).
  • Four tasks (as mentioned in the talk) for constructing the similar-triangle setup:
    • Task 1: Identify the two rays from the light source that just graze the edges of the opaque object.
    • Task 2: Extend those rays to hit the screen to locate the shadow boundaries.
    • Task 3: Observe that the triangles formed by the light source, the object edge, and the projection on the screen are similar.
    • Task 4: Use the similarity to relate heights and distances, determining where the screen is illuminated or in shadow.
  • Core takeaway: the projection from the object to the screen via the light source is governed by similarity of the triangles formed by the light-rays, the object, and the screen.

Setup: Two Light Sources, Opaque Object, and Screen Locations

  • The setup under discussion includes two light sources, an opaque object, and two possible screen locations (one labeled as Screen Two).
  • Screen Two is placed in a position where the projection of light from the sources can be analyzed with the same geometric setup as in the single-source case.
  • With two light sources, we can think of each source creating its own shadow on the screen, and the combination determines the regions of illumination.
  • On Screen Two, there is a region of full light where the red and blue rays both reach the screen (neither is blocked by the object).
  • There are regions where:
    • Red light is blocked (red shadow component) but blue light may still reach.
    • Blue light is blocked (blue shadow component) but red light may still reach.
  • The talk emphasizes that the two colored shadows (red and blue) are results of each light source being occluded differently by the same object, leading to distinct boundary lines on the screen.
  • The boundaries of these shadow regions are given by the lines from each light source that just graze the edges of the opaque object.
  • The description mentions a dynamic element: the camera (or observer) moving relative to the setup (e.g., "the camera was getting closer to the holes"), which changes the perceived size and location of the shadow regions on the screen.

Regions on the Screen: Full Light, Red-Blocked, Blue-Blocked

  • Region of full light: both light sources illuminate that part of the screen (neither red nor blue is blocked).
  • Red-blocked region: the red light is occluded by the object, so only blue light reaches that portion of the screen (assuming blue is not blocked there).
  • Blue-blocked region: the blue light is occluded by the object, so only red light reaches that portion of the screen (assuming red is not blocked there).
  • The arrangement implies a spatial separation of shadows corresponding to each light source, creating colored shadow regions when the light sources are colored (red, blue).
  • The geometry of these regions is governed by the same similar-triangle relationships as in the single-source case, but now applied separately to each light source.
  • The remark about the camera approaching the holes hints at how projection scales with distance: moving the observer (or screen) alters the apparent size of the shadow regions consistent with the similarity scaling.

Observations and Intuition

  • Parallel rays from distant sources produce predictable shadow boundaries through simple proportional reasoning (similar triangles).
  • The concept of two light sources illustrates how shadows can be composed: each source produces its own shadow region, and their overlap determines where full light occurs.
  • The boundary lines on the screen are the intersections of the screen with the rays that just graze the edges of the opaque object.
  • As the screen moves closer or farther from the light-object system, the size of shadow regions scales according to the geometric similarity described earlier.
  • The reference to "holes" and opening the other side suggests changes in the available light paths, which would shift or reveal different shadow boundaries due to the same geometric rules.

Connections to Foundational Principles

  • Similar triangles as a fundamental tool in optical projection: the shadow edges are found by tracing rays from the light source through the object to the screen; corresponding triangles are similar, yielding proportional relationships between heights and distances.
  • Pinhole-camera-like reasoning: the image on a screen is the result of projecting rays through a single or multiple light sources; the scale of the image is governed by the ratio of distances along the light path.
  • Occlusion and shadow boundaries arise from geometric constraints rather than color mixing alone; regions of illumination are determined by whether rays from each source reach the screen.

Practical Implications and Real-World Relevance

  • Understanding how shadows form with distant light sources helps in photography, stage lighting, and architectural shading design.
  • Analyzing multi-source lighting with occluders provides insight into color separation shadows (e.g., red and blue components) and how to plan lighting layouts to achieve desired effects.
  • The approach demonstrates why moving a screen or observer changes the observed shadow sizes—critical for calibrating projection systems or designing optical experiments.

Notation, Drawings, and How to Reproduce the Construction

  • To replicate the construction described:
    • Draw the opaque object and place one or more light sources at some distance from the object along the light path.
    • Choose a screen location where you want to observe the projection.
    • Draw the light rays that just graze the edges of the object; extend these rays to intersect the screen.
    • Mark the regions on the screen where each ray reaches; label regions of full light, red-blocked, and blue-blocked if using colored sources.
    • Use the fact that the triangles formed by the light source, the object edge, and the intersection on the screen are similar to relate the sizes and positions of the shadow regions.
  • The line l (the line of the light path) is a useful reference to illustrate the correspondence between object height, screen distance, and shadow height via similar triangles.

Quick Derivation Sketch (no numerical values provided in transcript)

  • Let the light source be at distance $Do$ from the object and the screen be at distance $Ds$ from the light along the same line (object lies between source and screen).
  • The shadow height on the screen is $h{ ext{image}}$ and the object height is $h{ ext{object}}$.
  • By similar triangles formed by the lines from the light source to the top of the object and to the top of the image on the screen, the scale factor is the ratio of distances from the light:

h<em>extimage = h</em>extobject ⋅ D<em>sD</em>o.h<em>{ ext{image}} \,=\, h</em>{ ext{object}} \,\cdot\, \frac{D<em>s}{D</em>o}.

  • A parallel-ray (sun-like) assumption corresponds to treating the rays as effectively parallel, in which case the same similarity principle applies to projections along the line of sight, and moving the screen changes the scale according to the same distance ratio.

  • If using two light sources with colors, apply the same derivation separately for each source to obtain two sets of boundaries on the screen; their overlap defines the regions of full illumination.

Summary of Takeaways

  • Parallel or near-parallel light sources yield shadow boundaries that can be analyzed with similar triangles.
  • A single opaque object in front of one or more light sources projects shadows on a screen; the boundaries of these shadows are determined by lines from the light sources that graze the object edges.
  • With two light sources, the screen shows regions of full light and color-specific shadows depending on which source is occluded, creating a composition of shadow regions.
  • The observed pattern can change with screen position or observer distance, but the underlying geometry remains governed by similar-triangle scaling.