bezier_curve_drawing_posts.txt

# Social Media Posts: Bézier Curve Drawing

## SHORT-FORM POSTS

### Twitter/X (280 chars)
Ever wonder how fonts and graphics stay smooth at any zoom? Bézier curves! Just a few control points create perfectly smooth paths using Bernstein polynomials: B(t) = Σ C(n,i)·tⁱ·(1-t)ⁿ⁻ⁱ·Pᵢ

#Python #Math #ComputerGraphics #Coding

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### Bluesky (300 chars)
Explored the mathematics behind Bézier curves today. These elegant parametric curves use Bernstein polynomials to interpolate control points: B(t) = Σ C(n,i)·tⁱ·(1-t)ⁿ⁻ⁱ·Pᵢ

From car body design to vector fonts, they're everywhere in graphics.

#Python #Mathematics

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### Threads (500 chars)
Just built a complete Bézier curve implementation from scratch!

These curves are everywhere - fonts, SVG graphics, animation paths, CAD design. The math is beautiful:

B(t) = Σ C(n,i)·tⁱ·(1-t)ⁿ⁻ⁱ·Pᵢ

Key insights:
• Curve always passes through first & last control points
• Stays inside the convex hull of control points
• De Casteljau's algorithm gives numerical stability
• Rational Bézier curves can draw perfect circles!

Drew a heart shape and decorative patterns to demo the power.

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### Mastodon (500 chars)
Implemented Bézier curves with both Bernstein polynomial evaluation and De Casteljau's recursive algorithm.

Bernstein form: Bᵢ,ₙ(t) = C(n,i)·tⁱ·(1-t)ⁿ⁻ⁱ

De Casteljau recursion: Pᵢ⁽ʳ⁾(t) = (1-t)·Pᵢ⁽ʳ⁻¹⁾ + t·Pᵢ₊₁⁽ʳ⁻¹⁾

Also explored rational Bézier curves - with weight w₁ = 1/√2, you get an exact circular arc! The notebook includes visualizations of control point manipulation and C¹ continuous splines.

#Python #Math #CompSci

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## LONG-FORM POSTS

### Reddit (r/learnpython or r/math)

**Title:** I built a complete Bézier curve implementation from scratch - here's how the math works

**Body:**

I just finished a deep dive into Bézier curves and wanted to share what I learned!

**What are Bézier curves?**

They're parametric curves used everywhere - fonts, vector graphics, CAD, animation. Named after Pierre Bézier who developed them for car body design at Renault.

**The core math (ELI5 version):**

Imagine you have a few "control points" - the curve smoothly flows from the first to the last, being "pulled" toward the middle ones. The formula uses Bernstein polynomials:

B(t) = Σ C(n,i)·tⁱ·(1-t)ⁿ⁻ⁱ·Pᵢ

Where t goes from 0 to 1, and C(n,i) is the binomial coefficient.

**Cool properties I discovered:**

1. **Endpoint interpolation** - curve passes through P₀ and Pₙ exactly
2. **Convex hull containment** - curve never leaves the "boundary" of control points
3. **Tangent behavior** - slope at endpoints matches direction to adjacent control point
4. **Affine invariance** - rotate/scale control points, curve transforms identically

**De Casteljau's algorithm:**

Instead of computing polynomials directly, you can recursively interpolate between points:

P⁽ʳ⁾(t) = (1-t)·P⁽ʳ⁻¹⁾ + t·P⁽ʳ⁻¹⁾

This is more numerically stable and shows the geometric construction beautifully.

**Rational Bézier curves:**

By adding weights to control points, you can draw exact conic sections. With weight 1/√2 on the middle control point of a quadratic curve, you get a perfect circular arc!

**What I built:**

- Bernstein polynomial evaluation
- De Casteljau's algorithm
- Rational Bézier curves
- Composite splines with C¹ continuity
- Fun shapes (heart, decorative patterns)

Check out the full notebook with code and visualizations:
https://cocalc.com/github/Ok-landscape/computational-pipeline/blob/main/notebooks/published/bezier_curve_drawing.ipynb

Happy to answer questions about the implementation!

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### Facebook (500 chars)

Ever wonder how computer graphics create perfectly smooth curves? Meet Bézier curves!

These mathematical gems use just a few control points to generate beautiful, smooth paths. They're behind everything from the fonts you're reading to car body designs.

I built an interactive notebook exploring the math and creating cool shapes - including a heart and decorative patterns!

The magic formula: B(t) = Σ C(n,i)·tⁱ·(1-t)ⁿ⁻ⁱ·Pᵢ

Check it out: https://cocalc.com/github/Ok-landscape/computational-pipeline/blob/main/notebooks/published/bezier_curve_drawing.ipynb

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### LinkedIn (1000 chars)

**Exploring the Mathematics Behind Smooth Curves in Computer Graphics**

Just completed a comprehensive implementation of Bézier curves, the mathematical foundation behind vector graphics, font rendering, and CAD systems.

**Technical Highlights:**

• Implemented Bernstein polynomial evaluation: Bᵢ,ₙ(t) = C(n,i)·tⁱ·(1-t)ⁿ⁻ⁱ

• Built De Casteljau's recursive algorithm for numerically stable curve evaluation

• Explored rational Bézier curves for exact conic section representation

• Created composite splines with C¹ continuity for complex shapes

**Key Skills Demonstrated:**

- Mathematical modeling and parametric curve theory
- NumPy/SciPy for scientific computing
- Matplotlib for technical visualization
- Algorithm implementation (recursive and iterative approaches)

**Applications:**

These curves are fundamental to graphics programming, game development, font design (TrueType, PostScript), and industrial CAD. Understanding the underlying mathematics enables better optimization and customization of graphics systems.

View the complete implementation with interactive visualizations:
https://cocalc.com/github/Ok-landscape/computational-pipeline/blob/main/notebooks/published/bezier_curve_drawing.ipynb

#Python #Mathematics #ComputerGraphics #DataVisualization #SoftwareDevelopment

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### Instagram (500 chars)

The art of smooth curves

These beautiful patterns were created using Bézier curves - the same math that powers fonts, vector graphics, and animation.

Just a few control points create perfectly smooth paths through Bernstein polynomials:

B(t) = Σ C(n,i)·tⁱ·(1-t)ⁿ⁻ⁱ·Pᵢ

The decorative spiral pattern uses 8 connected cubic Bézier segments.

Also drew a heart shape with just 2 curves and 8 control points!

Math really is beautiful when you visualize it.

#python #math #coding #dataviz #computerscience #graphics #visualization #matplotlib