How can a cubic Bezier curve be constructed to have a tangent to a circle?

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SUMMARY

A cubic Bezier curve can be constructed to have a tangent to a circle by ensuring that the control points of the Bezier curve are positioned correctly relative to the circle's radius and center. The first step involves defining the circle's parameters, followed by strategically placing the control points of the cubic Bezier curve to achieve the desired tangency. This construction requires precise calculations to ensure that the derivative of the Bezier curve at the point of tangency matches the slope of the circle's tangent line.

PREREQUISITES
  • Understanding of cubic Bezier curves and their control points
  • Knowledge of circle geometry, including tangents and radii
  • Familiarity with calculus concepts, particularly derivatives
  • Experience with graphical software or programming libraries that handle Bezier curves
NEXT STEPS
  • Study the mathematical properties of cubic Bezier curves in detail
  • Learn about circle tangents and their geometric implications
  • Explore software tools like Adobe Illustrator or programming libraries such as p5.js for practical applications
  • Investigate numerical methods for calculating derivatives of Bezier curves
USEFUL FOR

Graphic designers, computer graphics programmers, and mathematicians interested in curve design and geometric constructions will benefit from this discussion.

mymachine
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Can anyone describe a tangency of Bezier curve such as below image?
 

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This is a pretty vague request. Can you be more specific?
 
SteamKing said:
This is a pretty vague request. Can you be more specific?

The circle is construct first.

Then how construct a Bezier curve, say, a cubic Bezier curve, which has its tangent to the circle?

I hope you understand.
 

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