Curve Conversion

Fontisan handles conversion between TrueType's quadratic Bézier curves and OpenType/CFF's cubic Bézier curves.

Overview

Format	Curve Type	Math
TrueType (TTF)	Quadratic Bézier	1 control point
OpenType/CFF (OTF)	Cubic Bézier	2 control points

Quadratic Bézier Curves (TrueType)

Quadratic curves have one control point between two on-curve points:

P0 ---- Q1 ---- P2

Where:
- P0, P2 are on-curve points
- Q1 is the control point

Formula

B(t) = (1-t)²·P0 + 2(1-t)t·Q1 + t²·P2

Cubic Bézier Curves (OpenType/CFF)

Cubic curves have two control points:

P0 -- C1 -- C2 -- P3

Where:
- P0, P3 are on-curve points
- C1, C2 are control points

Formula

B(t) = (1-t)³·P0 + 3(1-t)²t·C1 + 3(1-t)t²·C2 + t³·P3

Quadratic → Cubic (TTF → OTF)

This conversion is mathematically exact.

Conversion Process

# TrueType quadratic: P0, Q1, P2
# Converts to cubic: P0, C1, C2, P2
#
# C1 = P0 + (2/3)(Q1 - P0)
# C2 = P2 + (2/3)(Q1 - P2)

Example

# Source: TrueType quadratic curve
p0 = [100, 100]
q1 = [150, 200]  # control point
p2 = [200, 100]

# Convert to cubic
c1 = [
  p0[0] + (2.0/3) * (q1[0] - p0[0]),  # 133.33
  p0[1] + (2.0/3) * (q1[1] - p0[1])   # 166.67
]
c2 = [
  p2[0] + (2.0/3) * (q1[0] - p2[0]),  # 166.67
  p2[1] + (2.0/3) * (q1[1] - p2[1])   # 166.67
]

# Result: [100, 100] -> [133.33, 166.67] -> [166.67, 166.67] -> [200, 100]

Characteristics

• Exact — No approximation needed
• Point count may increase — Due to implicit on-curve points in TrueType

Cubic → Quadratic (OTF → TTF)

This conversion requires approximation since cubic curves cannot be exactly represented as quadratic.

Approximation Methods

Fontisan uses the midpoint approximation method:

# Cubic: P0, C1, C2, P3
# Approximate with multiple quadratic curves
#
# Method: Subdivide until quadratic fits within tolerance

Tolerance Setting

options = Fontisan::ConversionOptions.new(
  curve_tolerance: 0.5  # Lower = more accurate, more points
)

Tolerance	Accuracy	Point Count
1.0	Lower	Fewer
0.5	Medium	Medium
0.1	Higher	More

Example

# Source: Cubic curve
p0 = [100, 100]
c1 = [120, 200]
c2 = [180, 200]
p3 = [200, 100]

# May require 2+ quadratic curves:
# Quad 1: P0 -> Q1 -> P_mid
# Quad 2: P_mid -> Q2 -> P3

Characteristics

• Approximate — Some precision loss unavoidable
• Point count increases — Multiple quadratic curves per cubic
• Tolerance affects quality — Lower tolerance = more accurate

Implicit On-Curve Points

TrueType uses an optimization where on-curve points can be implicit:

# If two consecutive off-curve points exist,
# the midpoint is an implicit on-curve point
#
# Q1 ---- (implicit P) ---- Q2
#
# P = (Q1 + Q2) / 2

This can reduce point count but must be expanded for conversion.

Curve Conversion Options

CLI

# Enable curve conversion
fontisan convert font.ttf --to otf --convert-curves

# With tolerance (OTF → TTF only)
fontisan convert font.otf --to ttf --curve-tolerance 0.5

API

options = Fontisan::ConversionOptions.new(
  from: :ttf,
  to: :otf,
  opening: {
    convert_curves: true
  },
  curve_tolerance: 0.5  # For OTF → TTF
)

Preserving Curves

To skip curve conversion (same-format operations):

options = Fontisan::ConversionOptions.new(
  opening: { convert_curves: false }
)

Quality vs Size Trade-off

TTF → OTF

• No trade-off — Exact conversion
• File size may increase due to explicit points

OTF → TTF

• Trade-off exists — Approximation required
• Lower tolerance = better quality, larger files
• Higher tolerance = lower quality, smaller files

# High quality
fontisan convert font.otf --to ttf --curve-tolerance 0.1

# Smaller file
fontisan convert font.otf --to ttf --curve-tolerance 1.0

Technical Details

Checking Curve Types

font = Fontisan::FontLoader.load('font.ttf')

# Check glyph format
glyph = font.glyphs[0]
puts glyph.curve_type  # :quadratic or :cubic

Counting Points

font = Fontisan::FontLoader.load('font.ttf')

total_on_curve = 0
total_off_curve = 0

font.glyphs.each do |glyph|
  glyph.contours.each do |contour|
    contour.points.each do |point|
      if point.on_curve?
        total_on_curve += 1
      else
        total_off_curve += 1
      end
    end
  end
end

puts "On-curve: #{total_on_curve}"
puts "Off-curve: #{total_off_curve}"