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Topic: Bézier curves

[WJGO]

Please explain the mathematical formula for a Quadratic Bézier Curve as used in drawing glyphs in fontmaking software.

[Copilot]
Ah, the Quadratic Bézier Curve—a foundational tool in digital typography and glyph design! Its formula is quite elegant and allows for smooth, precise curves, which are essential in fontmaking.

A Quadratic Bézier Curve is defined mathematically as:

[Note by WJGO P0, P1, P2 are meant to be P with a subscript 0, P with a subscript 1, P with a subscript 2. Subscripts are used in the original but not in this transcript.]

IS THE FORMULA THE ISSUE?

where:

  • Alfred
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Re: Bézier curves

William wrote:

[Note by WJGO P0, P1, P2 are meant to be P with a subscript 0, P with a subscript 1, P with a subscript 2. Subscripts are used in the original but not in this transcript.]

I’m having difficulty with those subscript numerals, too! This is the best I can do:

A quadratic Bézier curve is a parametric curve defined using three control points. The formula for the curve is:

B(t) = (1 − t)²P₀ + 2(1 − t)tP₁ + t²P₂

where

• P₀, P₁, and P₂ are the control points
• t is the parameter ranging from 0 to 1

This formula ensures a smooth curve that starts at P₀, moves toward P₁ as an intermediate guide, and ends at P₂. Quadratic Bézier curves are commonly used in computer graphics, font rendering, and animations.

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Re: Bézier curves

Thank you.

I had in mind asking Copilot again and doing a Print Screen and then adding a png image to this thread.

  • Alfred
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Re: Bézier curves

I get better results in a text editor, where the subscript numerals aren’t rendered as vanishingly small.
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“Welcome to the present moment. Here. Now. The only moment there ever is.”
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5 Reply by William (edited by William 2025-05-21 08:39:41)

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Re: Bézier curves

[Note on Wednesday. Copilot continued on Tuesday]

t is the parameter that varies from 0 to 1, tracing the curve.

This formula ensures that the curve starts at P0, is pulled toward P1, and finally reaches P2.

The influence of P1 on the curvature is strongest when t = 0.5, making it a critical component in defining how smooth or sharp the curve appears.

[The next paragraph from Copilot has been deleted as it was incorrect, as mentioned in a post later in this thread.]

  • Alfred
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Re: Bézier curves

William wrote:

Fontmaking software like TrueType and OpenType …

Huh?

“Welcome to the present moment. Here. Now. The only moment there ever is.”
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Re: Bézier curves

The problem with posting that I was having appears to be that the text produced by Copilot when copied from the web page and pasted into WordPad was typesetting the names of the variables in italics. I think it was using the special ISO/IEC 10646 / Unicode special mathematical characters. The software under the bonnet (under the hood in American English) in this forum was giving up when it met such a character.

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Re: Bézier curves

Alfred wrote:
William wrote:

Fontmaking software like TrueType and OpenType …

Huh?

Ah yes, I had not noticed that! Arrgh!

You know of course, but for completeness TrueType and OpenType are fontmaking format specifications. Fontmaking software applies those format specifications.

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Re: Bézier curves

William wrote:

The problem with posting that I was having appears to be that the text produced by Copilot when copied from the web page and pasted into WordPad was typesetting the names of the variables in italics. I think it was using the special ISO/IEC 10646 / Unicode special mathematical characters. The software under the bonnet (under the hood in American English) in this forum was giving up when it met such a character.

The text in Copilot itself uses italics for B, P, and t.

“Welcome to the present moment. Here. Now. The only moment there ever is.”
― Eckhart Tolle
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Re: Bézier curves

When I pasted into WordPad, the font Cambria Math became used automatically.

Normally WordPad (at least here) uses the Calibri font unless told otherwise.