Mass Curve Multiplier CalculationWIP

Published: • Updated:

This document explains the mathematical formula used to calculate mass curve multipliers for thrusters and shields in Elite: Dangerous. These multipliers determine how module performance scales with ship mass.

Overview

The mass curve multiplier formula is used by thrusters and shields to adjust their performance based on the ship’s mass. It creates a smooth curve that provides optimal performance at a specific mass range, with diminishing returns as the mass deviates from the optimal range.

Formula

The mass curve multiplier is calculated using the following formula:

\[\text{multiplier} = \text{minimumMultiplier} + \text{powerTerm} \times (\text{maximumMultiplier} - \text{minimumMultiplier})\]

Where the power term is:

\[\text{powerTerm} = \min\left(1.0, \frac{\text{maximumMass} - \text{mass}}{\text{maximumMass} - \text{minimumMass}}\right)^{\text{exponent}}\]

And the exponent is:

\[\text{exponent} = \frac{\ln\left(\frac{\text{optimalMultiplier} - \text{minimumMultiplier}}{\text{maximumMultiplier} - \text{minimumMultiplier}}\right)}{\ln\left(\frac{\text{maximumMass} - \text{optimalMass}}{\text{maximumMass} - \text{minimumMass}}\right)}\]

Parameters

The formula uses the following parameters from thruster and shield modules:

Formula Breakdown

Step 1: Mass Normalization

First, the mass is normalized to a value between 0 and 1:

\[\text{massRatio} = \frac{\text{maximumMass} - \text{mass}}{\text{maximumMass} - \text{minimumMass}}\]

This ratio is then clamped to a maximum of 1.0:

\[\text{clampedMassRatio} = \min(1.0, \text{massRatio})\]

Step 2: Exponent Calculation

The exponent determines the shape of the power curve. It’s calculated using logarithmic interpolation to ensure the curve passes through the optimal point:

\[\text{exponent} = \frac{\ln\left(\frac{\text{optimalMultiplier} - \text{minimumMultiplier}}{\text{maximumMultiplier} - \text{minimumMultiplier}}\right)}{\ln\left(\frac{\text{maximumMass} - \text{optimalMass}}{\text{maximumMass} - \text{minimumMass}}\right)}\]

Step 3: Power Term Calculation

The power term applies the exponential curve to the normalized mass:

\[\text{powerTerm} = \text{clampedMassRatio}^{\text{exponent}}\]

Step 4: Final Multiplier

Finally, the power term is scaled and offset to produce the final multiplier:

\[\text{multiplier} = \text{minimumMultiplier} + \text{powerTerm} \times (\text{maximumMultiplier} - \text{minimumMultiplier})\]

Visual representation

This is what the mass curve multplier looks like for a module with the following parameters:

{
	"$schema": "https://vega.github.io/schema/vega/v5.json",
	"width": 800,
	"height": 600,
	"title": {"text": "Mass curve multiplier for 7A Thrusters","font": "Inter, system-ui, sans-serif","fontSize": 20,"fontWeight": "bold","color": "#888","offset": 10},
	"data":{"name":"resistance_data","format":{"type":"csv"},"url":"mcm_values.csv"},
	"scales": [
		{"name": "xscale","type": "linear","domain": [500.0,4000.0],"zero": false,"range": "width"},
		{"name": "yscale","type": "linear","domain": [90.0,120.0],"zero": false,"range": "height"}
	],
	"axes": [
		{"scale": "xscale","orient": "bottom","title": "Ship mass","labelColor": "#888","titleColor": "#888","labelFontSize": 12,"titleFontSize": 14,"grid": true,"gridColor": "#888"},
		{"scale": "yscale","orient": "left","title": "Mass curve multiplier (%)","labelColor": "#888","titleColor": "#888","labelFontSize": 12,"titleFontSize": 14,"grid": true,"gridColor": "#888"}
	],
	"marks": [
		{"type": "line","from": {"data": "resistance_data"},"encode": {"enter": {"x": {"scale": "xscale","field": "x"},"y": {"scale": "yscale","field": "mcm"},"stroke": {"value": "orange"},"strokeWidth": {"value": 2}}}}
	]
}

Example Calculation

For a thruster module with the following parameters:

Step 1 - Mass Normalization:

\(\text{massRatio} = \frac{800 - 400}{800 - 100} = \frac{400}{700} \approx 0.571\) \(\text{clampedMassRatio} = \min(1.0, 0.571) = 0.571\)

Step 2 - Exponent Calculation:

\[\text{exponent} = \frac{\ln\left(\frac{1.0 - 0.5}{1.2 - 0.5}\right)}{\ln\left(\frac{800 - 200}{800 - 100}\right)} = \frac{\ln\left(\frac{0.5}{0.7}\right)}{\ln\left(\frac{600}{700}\right)} = \frac{\ln(0.714)}{\ln(0.857)} \approx \frac{-0.336}{-0.154} \approx 2.18\]

Step 3 - Power Term:

\[\text{powerTerm} = 0.571^{2.18} \approx 0.295\]

Step 4 - Final Multiplier:

\[\text{multiplier} = 0.5 + 0.295 \times (1.2 - 0.5) = 0.5 + 0.295 \times 0.7 \approx 0.5 + 0.206 = 0.706\]

Key Properties

  1. Optimal Performance: The curve is designed to pass through the optimal multiplier at the optimal mass
  2. Bounded Output: The result is always between the minimum and maximum multipliers
  3. Smooth Transition: The logarithmic exponent ensures smooth transitions between mass ranges
  4. Mass Clamping: When mass is below minimumMass, the multiplier equals maximumMultiplier
  5. Diminishing Returns: Performance degrades more as mass increases

Mathematical Behavior