Heart Curve Visualization

Explore the Romance of Mathematics - Parametric Heart Curve

Parametric Equations

x = 16 sin³(t)
y = 13 cos(t) - 5 cos(2t) - 2 cos(3t) - cos(4t)

Current Parameters

t = 0.00 rad
x = 0.00
y = 16.00

Drawing Progress

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What is a Heart Curve?

The Heart Curve is a mathematical curve shaped like a heart, combining romance and mathematics perfectly. It has a long history in mathematics and is often used to express love and create Valentine's Day themed art and design.

Parametric Equations

The most commonly used parametric equations for the heart curve are:

x = 16 sin³(t)
y = 13 cos(t) - 5 cos(2t) - 2 cos(3t) - cos(4t)

where parameter t ranges from 0 to 2π.

Implicit Equation

The heart curve can also be expressed as an implicit equation:

(x² + y² - 1)³ - x²y³ = 0

Mathematical Properties

  • Symmetry: The heart curve is symmetric about the y-axis, matching the natural shape of a heart.
  • Cusp: The curve has a cusp at the bottom, corresponding to t = 3π/2.
  • Smoothness: Except for the cusp, the curve is smooth everywhere else.
  • Area: The area enclosed by the standard heart curve is approximately 180.9 square units.

Applications

  • Art & Design: Heart patterns are widely used in card design, jewelry design, and tattoo art.
  • Math Education: Used to demonstrate parametric equations and trigonometric functions.
  • Computer Graphics: Used to generate decorative graphics and animation effects.
  • Physics: Similar shapes appear in certain wave and optics problems.

Historical Background

The mathematical study of heart curves dates back to the 17th century, closely related to classical curves such as cycloids and rose curves. Although the mathematical description of heart curves has existed for a long time, their popularity as a "love" symbol began only in modern times. This curve perfectly combines mathematical precision with human emotional expression, becoming a classic example of mathematical beauty.

Heart Curve Variants

By adjusting parameters, different styles of heart curves can be obtained:

  • Classic Heart: Using the standard parametric equations above.
  • Algebraic Heart: Using the implicit equation (x² + y² - 1)³ - x²y³ = 0.
  • Polar Heart: r = 1 - sin(θ), which is a different type of heart curve.