Elevation & Azimuth

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Standard photography typically employs conventional lenses devoid of artificial effects, a factor that simplifies the estimation of the sun's position at the moment of capture. Nevertheless, achieving a rigorous analytical evaluation requires specific technical conditions. This section outlines the fundamental principles governing this methodology.

1. Preliminary Photographic Principles

2. Methodology: Estimating Solar Angular Position via Horizontal Spherical Coordinates

3. Computational Analysis and Simulation Results

1. Preliminary Photographic Principles

Prior to a detailed analysis, it is essential to establish several fundamental photographic concepts. To illustrate these, we utilize a Google Maps® screen capture characterized by a wide Field of View (FOV).

Wide-angle photo generated by Google Maps® from a 360° panoramic photograph.

A wide-angle perspective facilitates the identification of the three primary vanishing points (VPs)—coordinates where parallel lines, as perceived through a specific perspective, converge toward an infinite distance:

Solar Vanishing Point (VP): From a terrestrial perspective, solar rays function as nearly parallel lines of light. By analyzing the shadows cast by the truck, we can precisely locate the solar vanishing point.
Road Vanishing Point (rVP): The receding parallel lines of the roadway converge at the rVP, situated along the estimated Horizon Line.
Vertical Vanishing Point (vVP): The vertical orientation of the lamp posts serves to identify the estimated vertical vanishing point (vVP).

Technical Analysis of the Google Maps® Projection.

Geometric Parameters:

Vanishing Points: Solar (VP), Vertical (vVP), and Road (rVP).
Optical Reference: Center of View (C), Horizon Line, and Field of View (FOV).

Analysis:

The camera orientation is rotated approximately 50° relative to the direction of traffic. This image utilizes a planar (rectilinear) projection, standard in conventional photography. In such projections, the FOV remains strictly below 180°, as a 180° field would necessitate an infinitely wide image plane.
Note the pronounced geometric distortions regarding both angles and distances as the perspective moves away from the Center of View (C). The estimated FOV is approximately 144°, which translates to a 35mm equivalent focal length (EFV) of roughly 5.85 mm. This ultra-wide rectilinear projection is characteristic of the immersive panoramic imagery provided by Google Maps®. While such a focal length induces significant geometric compression at the periphery, it remains a robust baseline for triangulating the solar position relative to the local horizon.
While the vVP is located outside the frame in this instance, it remains a critical coordinate for orientation. Note that when the sun is positioned behind the camera, light rays converge toward an anti-solar vanishing point (as demonstrated in the "Methodology" section below).

Geometric Constraints

The angular distance between the Horizon Line and the vertical vanishing point (vVP) is exactly 90°. In this specific capture, the truck’s cab and the primary lamp post are oriented nearly perpendicular to the roadway. This orthogonality is confirmed by the alignment of the dual lamps at the post's summit and the truck's perspective (verified by a 90° secondary view at the same coordinates). This configuration establishes a 90° reference point on the Horizon Line, yielding an estimated FOV of approximately 145° (*).

(*) Methodological Limitations:

This image serves as a preliminary estimation rather than a definitive measurement. Any rigorous evaluation requires a comprehensive contextual analysis prior to calculation, as extreme wide-angle projections are inherently sensitive to minor margins of error. Factors such as the road's slight downward incline (positioning the true Horizon Line marginally higher) and potential interpolation artifacts—stemming from the conversion of a spherical panoramic shot into a planar image—may introduce slight shifts in the Center of View (C) or the perceived verticality of the lamp posts.

Subsequent simulations (see below) suggest a corrected FOV of approximately 148°, with the vVP and Horizon Line positioned slightly higher than initially estimated.

Perspective oriented 90° counter-clockwise relative to the traffic flow, captured at the same temporal and geospatial coordinates.

2. Methodology: Estimating Solar Angular Position via Horizontal Spherical Coordinates

Note: The reference image of this example was rendered in Blender® using a 20 mm focal length, with a solar elevation of 30° and an azimuth of 140° relative to the viewing axis.

To determine the sun's position with analytical precision, we follow a structured six-step methodology:

1) Image Acquisition and Metadata Retrieval

Obtain a high-quality photograph characterized by sharp shadows, clearly defined vertical/horizontal planes, and discernible vanishing directions. Accessing internal camera metadata—such as focal length, Field of View (FOV), and pitch/roll angles—is essential.
Note: Prefer rectilinear lenses over specialized optics (e.g., fisheye) to avoid complex geometric distortions.

2) Identifying the Solar Vanishing Point (VP)

The VP is extrapolated from reflections, cast shadows, or shading. While an on-site light probe provides the ideal reference, shadows cast by long, distinct objects offer the most reliable terrestrial alternative. If the sun is within the frame, the VP is its geometric center; if positioned behind the camera, we utilize the anti-solar point (as seen in this example).

3) Establishing the Horizon Line (H)

For a leveled camera, the Horizon Line remains parallel to the horizontal axis. Vertical structures help calibrate the "roll" angle (lateral tilt). If the camera exhibits a "pitch" angle (forward/backward tilt), vertical lines will converge toward a distant vanishing point (e.g., in the figure, verticals converge toward a point below the frame).

4) Locating the Center of View (C)

It is the point in the photo where the camera is pointing to. Normally it is located at the center of the photograph if this has not been cropped, and if the lens is centered without artifacts (as with shift or tilt lenses). With a zero pitch angle, C is located on the Horizon line H. A forward tilt gives C below H as in the figure.

5) Calibrating Focal Distance (FD) and Field of View (FOV)

The Focal Distance determines the FOV and dictates the projection's scale. By aligning the positions of H and C, we superimpose a spherical coordinate grid centered on the Horizon Center (Hc). Adjust the FD until the projected vertical grid aligns with the image's physical verticals (e.g., lamp posts).

Formula: FD = 18/tan(FOV/2). In our example, a horizontal span of 84° at C (-42° to +42°) confirms an FD of 20 mm.

6) Determining Solar Elevation φ and Relative Azimuth (AZ)

The final solar coordinates are derived from the angular abscissa and ordinate of the VP. When the sun is behind the observer, we calculate the inverse coordinates.

Example values: Pitch α =-7°; Elevation φ=30°; Azimuth AZ=140° (180° - 40° per Blender parameters; see the Blender Implementation).

Constraint: φ ≤ γ0(here γ0 ~42.3°). The true elevation φ is always less than or equal to the apparent elevation γ0 at the Center of View (Hc). Refer to the Mathematical Appendix for a detailed proof.

3. Computational Analysis and Simulation Results

By implementing these geometric principles within a 3D Blender® environment, we developed a universal Template applicable to any standard photograph. This allows for a rigorous graphical solution by superimposing a spherical coordinate system onto the 2D image.

Key Findings:

Solar Coordinates: The estimated solar elevation is approximately 52°, with a relative azimuth of 20°.
Image Geometry: The capture of Blender's template is cropped from approximately -68° to +61° along the active horizon line (cropped FOV ~ 129°).
Optical Parameters: The final calibration yields an equivalent focal length (FD) of ~5.15 mm, corresponding to a total Field of View (FOV) of ~148° (uncropped).

Graphical solution applied to the Google Maps® capture. The calibrated positions of the Horizon Line, the Vertical Vanishing Point (vVP), and the Center of View (C) were derived by adjusting the Focal Distance (FD) until the spherical grid aligned precisely with the physical orientation of the lamp posts.

© Ph. Dalleur

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