“Logarithmic Spiral Evolute”的概念、定义、习题解题方法及翻译-数学辞典

单词

Logarithmic Spiral Evolute

释义

Logarithmic Spiral Evolute

(1)

Using

(2)

gives

and


			(5)

and the Tangent Vector is given by


			(6)

The coordinates of the Evolute are therefore

			(7)
			(8)

So the Evolute is another logarithmic spiral with

, as first shown by Johann Bernoulli.

However, in some cases, the Evolute is identical to the original, as can bedemonstrated by making the substitution to the new variable

(9)

Then the above equations become

which are equivalent to the form of the original equation if

(12)

(13)

(14)

where only solutions with the minus sign in

exist. Solving gives the values summarized in the following table.

References

Lauwerier, H. Fractals: Endlessly Repeated Geometric Figures. Princeton, NJ: Princeton University Press, pp. 60-64, 1991.

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