01
Coordinates
Public road geometry is read as ordered coordinate points.
Maths Deep Dive
Maths Deep Dive for the Model Output
This page explains the public-facing formula chain. It does not change the live app model; it explains how road geometry, speed, friction, and comparative-output interpretation are connected.
Model Formula Chain
Coordinates to Comparative Output
02
Heading Change
Direction changes across the selected segment are measured.
03
Radius and Curvature
Segment length and heading change are converted into bend severity.
04
Vehicle-Dynamics Checks
Centripetal demand, safe speed, slip ratio, and stopping distance are estimated.
05
Scenario Modifiers
Weather, visibility, driver, surface, vehicle, and context assumptions are applied.
06
Percentile Context
The output is compared against sampled roads under the same assumptions.
Method Notes
Distinguishing Physical Relationships from Model Interpretation
Some values are derived directly from geometry. Some are physical checks based on standard motion relationships. Some are scenario modifiers or interpretive conversions that help the output make sense to users.
The important discipline is to keep those categories separate. A formula can be physically valid while the final risk interpretation still depends on assumptions, data quality, and calibration limits.
Model Equations
Core Mathematical Relationships
Heading Change
Directional Change Across the Segment
\[ \theta = \left|\beta_{\text{end}} - \beta_{\text{start}}\right| \]
- theta: absolute heading change across the analysed road segment.
- bearing: direction of travel between coordinate pairs.
Physical meaning: the model estimates directional change before deriving curvature and radius.
Appears in: curvature, radius, selected-road geometry, maths output.
Limitation: tiny heading changes on very short or noisy geometry can exaggerate curvature if not handled cautiously.
Radius of Curvature
Curve Tightness from Segment Geometry
\[ r = \frac{L}{\theta} \]
- r: radius of curvature in metres.
- L: analysed segment length in metres.
- theta: heading change in radians.
Road Risk uses this to translate map geometry into a physical bend radius.
Appears in: Safe Speed, curvature score, Annualised Comparative Model Output.
Limitation: very short, noisy, or sparsely mapped geometry can make radius estimates unstable.
Centripetal Acceleration
Lateral Acceleration Demand
\[ a_c = \frac{v^2}{r} \]
- a_c: centripetal acceleration.
- v: vehicle speed in metres per second.
- r: curve radius in metres.
A tighter bend or higher speed increases the lateral acceleration required to follow the curve.
Appears in: slip ratio, safe-speed interpretation, curvature-related risk context.
Limitation: real handling also depends on tyres, road banking, surface texture, driver response, and vehicle state.
Friction-Limited Safe Speed
Curve Speed Under a Friction Assumption
\[ v_{\text{safe}} = \sqrt{\mu_{\text{eff}} g r} \]
- mu_eff: effective friction coefficient after surface or scenario assumptions.
- g: gravitational acceleration.
- r: curve radius.
Road Risk uses this as a readable physical check against the active scenario speed.
Appears in: Safe Speed.
Limitation: this is a simplified friction threshold, not a full vehicle-dynamics simulation.
Slip Ratio
Demand Relative to Available Friction
\[ S = \frac{a_c}{\mu_{\text{eff}} g} \]
- a_c: lateral acceleration demand.
- mu_eff * g: simplified available friction acceleration.
A higher value means the scenario is closer to the assumed grip limit.
Appears in: Annualised Comparative Model Output and maths diagnostics.
Limitation: the ratio is a warning-style indicator, not a measurement of tyre slip from sensors.
Stopping Distance
Reaction and Braking Distance
\[ d_{\text{stop}} = v t_r + \frac{v^2}{2 \mu g} \]
\[ d_{\text{scaled}} = d_{80}\left(\frac{v}{22.22}\right)^2\left(\frac{\mu_0}{\mu_{\text{eff}}}\right)M_{\text{scenario}} \]
- v: speed in metres per second.
- t_r: reaction time assumption.
- mu_eff: effective friction coefficient after scenario adjustment.
- d_80: base braking-distance reference at 80 km/h.
- M_scenario: combined scenario modifier.
Road Risk uses this to explain how speed, friction, and scenario assumptions affect stopping demand. The velocity-squared term is why speed changes can dominate braking distance.
Appears in: Stopping Distance.
Limitation: gradients, ABS behaviour, tyre condition, visibility, and driver state are simplified.
Annual-to-Daily Conversion
Daily Interpretive Transformation
\[ p_{\text{daily}} = 1 - (1 - p_{\text{annual}})^{1/365} \]
- P_year: annualised comparative model output.
- P_day: equivalent daily interpretive view.
The daily view helps users read a small annual value without changing the primary comparison basis.
Appears in: Daily Interpretive View.
Limitation: this is an interpretive conversion of the model output, not an observed daily crash rate.
Scenario Multiplier Composition
Scenario-Adjusted Model Output
\[ R_{\text{scenario}} = R_{\text{base}} \times M_{\text{weather}} \times M_{\text{visibility}} \times M_{\text{driver}} \times M_{\text{surface}} \]
- M_total: combined scenario multiplier.
- M_weather, M_visibility, M_driver, M_vehicle, M_context: model assumption families.
Model use: scenario assumptions adjust the comparative output so sensitivity to conditions can be examined.
Appears in: Annualised Comparative Model Output, Daily Interpretive View, scenario summary.
Limitation: multiplier values are modelling assumptions and require careful validation before stronger claims are made.
Illustrative Worked Example
Illustrative Connection Between Geometry and Output
Illustrative Example
Selected Bend Under a Baseline Dry-Road Scenario
If the selected geometry has a short segment length and a noticeable heading change, the calculated radius becomes smaller. At the same scenario speed, smaller radius increases centripetal acceleration, which can reduce safe-speed margin and raise the comparative model output.
Not a measured case
This example explains the formula chain. It is not an exported real-road result and should not be interpreted as observed crash evidence.
Calculation Chain
Linking Geometry, Physics, and Model Interpretation
01
Geometry
Selected road coordinates are converted into segment length, heading change, radius, and curvature.
02
Physics-Informed Checks
Speed, radius, friction, reaction time, safe speed, and stopping distance are evaluated.
03
Scenario Multipliers
Weather, visibility, traffic, road context, driver behaviour, and vehicle profile affect interpretation.
04
Comparative Output
The app displays risk card values, maths, graph data, route analysis, and exports from the current context.
Formula-page limitation
This page explains the public method at a readable level. The live app contains additional implementation detail for fallback assumptions, route sampling, percentile context, exports, and UI state.
Live Application
Compare the Equations with Selected-Road Values
The app turns these relationships into selected-road outputs, scenario controls, graphs, route analysis, and exports.