Line of sight and Fresnel zone: check that a link gets through, on Québec’s relief

An antenna on a summit, a subscriber at the bottom of a valley: will the signal get through? A radio relay on a hill, an emergency crew across the lake: will they see each other? Drawing a straight line on a map doesn't answer the question. Between the two points lie terrain, forest, buildings — and the curvature of the Earth itself. Québec 3D's new Line of sight tool runs that check on 1-metre LiDAR relief, free, in your browser.

It extends the elevation profile: the same terrain cross-section, but between two points that each have a height — a mast, an antenna, an observer — and with one precise question: does the straight line between them clear everything that stands in the way?

Check a line of sight in four steps

1. In the map panel, click “Line of sight”.
2. Enter the height above ground of A and B, in metres: 30 m for a tower, 10 m for an antenna on a pole, 1.7 m for a standing observer. Optional: a frequency in MHz, to check the Fresnel zone.
3. Click on the map to set point A, then point B (on a phone, tap the map).
4. Read the verdict, then hover over the cross-section to read the clearance at each point. The CSV button exports the details.

Heights and frequency are remembered: from one check to the next, you only place the points.

The Earth is round, and the air bends radio waves

Over a short distance, the Earth looks flat. Over a link several kilometres long, it no longer is: the ground “bulges” between the two ends and rises toward the straight line. This Earth bulge, at a distance d₁ from A and d₂ from B, is:

b = d₁ × d₂ / (2 × k × R)

where R is the Earth's radius (~6,371 km). The coefficient k captures atmospheric refraction: the air, denser near the ground, bends radio waves slightly toward the Earth, as if it were larger. The standard value for radio links is k = 4/3; it is the one the tool uses.

The numbers are surprising: at the midpoint of a 10 km link, the bulge reaches about 1.5 m; at 20 km, nearly 6 m; at 40 km, more than 23 m — the height of a seven-storey building. On the chart, the tool keeps the terrain's true elevations and instead bends the line of sight down by that amount: it is geometrically equivalent, and the cross-section stays readable.

Forest matters as much as terrain

Judging a line of sight on bare earth would be too optimistic: in Québec, a mature stand routinely exceeds 20 to 25 metres, and a wooded ridge hides far more than a bare one. By default, the tool therefore judges the line against the surface — the ground plus the height of the canopy and buildings, measured by the same LiDAR. Untick “Surface” to compare with bare earth: the gap between the two verdicts tells you, on its own, how much the vegetation weighs on the link.

The heights of A and B are measured from the ground, not from the treetops: a 10 m antenna planted in a 20 m forest is, rightly, blinded from its base.

The Fresnel zone: a tunnel, not a thread

A radio wave doesn't travel along an infinitely thin thread. The useful energy flows through a cigar-shaped volume around the line A→B: the first Fresnel zone. Its radius, largest at the middle of the link, is:

r₁ = √( λ × d₁ × d₂ / D )

where λ is the wavelength (λ = c / f) and D the length of the link. The lower the frequency, the wider the zone: at the midpoint of a 10 km link, r₁ is about 33 m at 700 MHz, but 15 m at 3.5 GHz.

An obstacle that bites into this zone without cutting the line does not block the view, but it weakens the signal through diffraction. The rule of thumb is to keep at least 60% of the first zone's radius clear: at that threshold the obstacle loss becomes negligible, whereas a line that merely grazes an obstacle already loses around 6 dB. If you enter a frequency, the tool draws this zone in blue on the cross-section and computes the clear percentage at the worst point.

Reading the verdict

• Green — Clear line of sight: the line clears everything and, if a frequency is given, at least 60% of the Fresnel zone is free.
• Orange — Visible, Fresnel zone obstructed: the view is direct, but an obstacle bites into the Fresnel zone; the link will work, weakened.
• Red — Line of sight blocked: something cuts the line. The first obstacle is marked on the map, and the stretches of terrain or forest that pierce it are drawn in red on the cross-section.

The minimum clearance gives, in metres, the margin at the tightest point — or how far the obstacle exceeds the line. It is the number to watch: it gives the order of magnitude of the extra antenna height needed to clear the obstacle. The CSV file gives, point by point, the ground and surface elevations, the height of the line of sight, the clearance and the Fresnel radius.

What it is good for

Cellular telecommunications — Spot, before any study, whether a candidate site “sees” a road, a village or a sector to cover; compare several tower locations; estimate the antenna height that clears a wooded ridge; check a microwave backhaul link between two sites.

Radio, IoT and private networks — VHF/UHF relays for emergency services, LoRa gateways, point-to-point links for a farm or a municipality: the same clearance question arises, at other frequencies.

Landscape and planning — Will a wind turbine, a tower or a tall building be visible from a lookout, a village, a shoreline? The line of sight answers point by point.

Outdoors and curiosity — Can you see the river from this summit? The lighthouse from this cottage? With an observer height of 1.7 m, the tool will tell you.

Limits worth knowing

This is a first screening, not a propagation study. The tool tells you whether the path is geometrically clear; it does not compute diffraction loss, vegetation attenuation or reflections. The surface is smoothed: an isolated tree or a roof ridge may be slightly underestimated, and towers or power lines are not in it. The forest cover dates from the LiDAR flight: later cutting or regrowth is not reflected. Finally, k = 4/3 describes an average atmosphere; some conditions (temperature inversions) change the apparent curvature. To size a link, these results call for a proper radio study.

The takeaway

Two points see each other if the line joining them clears the terrain and what stands on it, once the Earth is curved. A radio wave needs more: a tunnel — the Fresnel zone — clear to roughly two thirds. On Québec's LiDAR relief, these checks now take two clicks.