# Deltoid

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## History

Conceived by Euler in 1745 in connection with a study of caustics curves.

## Description

Deltoid (aka tricuspid) is a special case of hypocycloid. (See: Curve Family Index)

Deltoid can be defined as the trace of a point on a circle, rolling inside another circle 3 or 3/2 times as large in radius. The latter is called double generation.

The two sizes of rolling circles that generate the deltoid can be synchronized by a linkage. (this means: the 2 roulette methods trace the curve with the same speed and has a geometric relation) Let A be the center of the fixed circle. Let D be the center of the smaller rolling circle. Let F be a fixed point on this circle (the tracing point). Let G be a point translated from A by the vector DF. G is the center of the large rolling circle, with the same tracing point at F. ADFG is a parallelogram with sides having constant lengths.

## Formula

- Parametric: {(2 Cos[t]) + Cos[2 t], (2 Sin[t]) - Sin[2 t]}, 0 < t ≤ 2 π.
- Cartesian: (x^2+y^2)^2-8 x (x^2 - 3 y^2) + 18 (x^2 + y^2) - 27 == 0

## Properties

### Curve Construction

The deltoid is rich in properties. Its locus, tangent, and center of osculating circle can be constructed. If given a segment of unit 1 with 1/3 marked, then the curve can be constructed with ruler and compass.

Let there be a circle c centered on O passing B. We will construct the curve centered on O with one cusp at B. Let O be the origin, and B be the point {1,0}. Let J be a variable point on c. Construct a point E on c such that angle[B,O,E] == - 2*angle[B,O,J]. Construct a circle d, centered on O with radius 1/3 of circle c. The line OE intersect d at 2 places. Let A be the intersection of d and segment OE. Let G be the (other) intersection of d and line OE. Let P be a point on line JE such that GP and JE are perpendicular. Now, the point P is the locus of deltoid as J varies. GP is its tangent. JE is its normal.

To construct the osculating circle: Let k be a circle with radius 2/3 of c. Let H be the intersection of k and OJ. Let Q be the mirror of P thru H. The intersection of OQ and EJ is the center of osculating circle at P.

Further, H is the center of the smaller rolling circle with tracing point at P, and A is the center of the larger rolling circle with tracing point at P. Points OHPA is a parallelogram with constant sides.

### Tangent

Let A be the center of the curve. Let B be one cusp, P be any point on the curve. Let E, H be the intersections of the curve and the tangent at P. The segment EH has constant length distance[E,H]== 4/3*distance[A,B]. The locus of midpoint D of tangent segment EH is the inscribed circle. The normals at E,P,H are concurrent, and its locus is the circumscribed circle. Let J be the intersection of another tangent cutting EH at right angle. The locus of J (deltoid's orthoptic) is the inscribed circle.

### Deltoid and Astroid

Astroid is the caustic of deltoid with parallel rays in any direction.

prove the caustic relation between deltoid and astroid.

### Evolute

put proof that the deltoid evolute is a deltoid.

The evolute of deltoid is another deltoid. (all epi/hypocycloids' evolute are equal to themselves) In the left figure, the evolute is formed by the envelope of its normals. The right figure show osculating circles and their centers.

### Inversion

A inversion of deltoid with respect to its center.

### Simson Lines

prove that deltoid is envelope of Simson lines

Deltoid is the envelope of Simson lines of any triangle. (Robert Simson, 1687 – 1768). Step by step description:

- Let there be a triangle inscribed in a circle.
- Pick any point P on the circle.
- Mark a point Q1 on any side of the triangle such that line[P,Q1] is perpendicular to it. Extend the side if necessary.
- Similarly, find points Q2 and Q3 with respect to P for other sides.
- The points Q1, Q2, and Q3 are colinear. The line passing through them is called Simson line of the triangle with respect to P.
- Find Simson lines for other points P on the circle. The envelope of Simson lines is the deltoid. Amazingly, this is true for
*any*triangle.

### Pedal, Radial, and Rose

prove deltoid's radial, pedal, rose curves

The pedal of deltoid with respect to a cusp, vertex, or center is a folium with one, two, or three loops respectively. The last one is called trifolium, which is a 3-petaled rose. Deltoid's radial is a trifolium too. (all epi/hypocycloid's pedal and radial are equal, and they are roses.) Conjectures: ① 60 degree isoptic of deltoid is a inscribed trifolium. visual proof ② The derivative of deltoid's parametric equation is also a trifolium.

Deltoid as Roulette Constructing Tangent Circle

## Related Web Sites

See: Websites on Plane Curves, Plane Curves Books.

Robert Yates: Curves and Their Properties.

The MacTutor History of Mathematics archive

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