Evolute and Envelope of Curves

Questions and Answers

What is the equation of the normal line to the hyperbola derived in the content?

(y - b sinh t) = -(b/a)(x - a cosh t)

y = (b/a)(x - a cosh t) + b sinh t

(y - b sinh t) = (b cosh t)(x - a cosh t)

(y - b sinh t) = -(a/b cosh t)(x - a cosh t) (correct)

The expression for tanh t derived in the content is positive.

False

Identify the variables used to represent the coordinates of any point on the hyperbola.

a cosh t and b sinh t

The variable ___ represents the relationship between sinh t and cosh t in the context of the hyperbola.

tanh t Signup and view all the answers

Match the following terms with their definitions:

Normal Line = A line perpendicular to the tangent of a curve at a given point Hyperbola = A type of conic section characterized by its two branches Cosh t = The hyperbolic cosine function Sinh t = The hyperbolic sine function Signup and view all the answers

What does the variable 'h' represent in the context of the discussion on the evolute of the hyperbola?

(ax)^2/3 - (by)^2/3 Signup and view all the answers

Differentiating the equation partially with respect to t provides crucial information for finding the evolute.

True Signup and view all the answers

State the hyperbola equation used in this context.

a^2 - b^2 = 1 Signup and view all the answers

In the equation (by)^13 + (ax)^13 = a^2 + b^2, __________ represents the values from the hyperbola.

h Signup and view all the answers

What does the factor 2/3 in the derived expressions indicate?

The power to which the variable is raised Signup and view all the answers

Study Notes

Evolute of Curves

The evolute refers to the locus of the centers of curvature of a given curve.
An evolute can represent a circle in specific cases.

Envelope Definition

An envelope is a curve that is tangent to each member of a family of curves.
For families of curves, there are defined procedures to find the envelope.

Procedure for Finding Envelope

Case 1: One Parameter Family of Curves

Consider curves described by ( y = f(x, α) ).
Step 1: Partially differentiate with respect to the parameter ( α ) and solve for ( α ).
Step 2: Substitute the value of ( α ) back into the original family equation to find the envelope.
Special Case: If the curve is quadratic in ( α ), the envelope is identified by setting the discriminant to zero ( B² - 4AC = 0 ).

Case 2: Two Parameter Family of Curves

Consider curves represented as ( y = f(x, α, β) ), connected by a relation ( g(α, β) = 0 ).
Step 1: Differentiate both equations with respect to ( α ) while treating ( β ) as dependent.
Step 2: Eliminate ( α ) and ( β ) to obtain the envelope.

Example Problems

Envelope of Straight Lines:
- Find the envelope of ( y = mx + am p ).
- Differentiate to express parameters, substitute to eliminate, and derive the envelope equation.
Envelope of Circular Family:
- For the equation ( x \sin θ - y \cos θ = aθ ), differentiate and solve for ( x ) and ( y ), leading to an expression for ( y ).
Leibnitz’s Problem:
- Calculate the envelope of circles centered on the x-axis, radii proportional to the x-coordinate. Resulting envelope can be summarized as ( 4xy = 1 ).
Family of Straight Lines:
- Given straight-line equations connected by a linear relation, derive the envelope ( x + y = 1 ).

Circle Equation for Generating Family

The equation for circles passing through the origin with centers at ( (α, β) ) is given by ( x² + y² - 2αx - 2βy = 0 ).

Differentiation for Normal Lines

The normal line to a hyperbola can be defined, providing relationships that allow the extraction of coordinates based on derivatives.

Result of Evolute for Hyperbola

The derived relationships through differentiation offer insights into the evolute of the hyperbola ( a²b² = 1 ), leading to specific parametric results.

Conclusion

Understanding envelopes and evolutes offers fundamental insight into geometry and curvature, enriching knowledge in calculus and differential equations.

Description

This quiz explores the concepts of evolutes and envelopes in curves. It includes definitions, procedures for finding envelopes, and distinguishes between one and two parameter families of curves. Test your understanding of these mathematical concepts and their applications.