Now we extend these constructions to oval forms.
'Similar operations using curves (circular, elliptic or parabolic) together with or without straight lines.---/'
The exposition continues with the extension of such straight lines to include curves, beginning with circular forms and parabolas and proceeding to vortices with or without the imposition of straight lines. The ways of forming these figures are given in a constructive manner rather than formally algebraically.

Fig.78. A circle is constructed by means of a pen attached to a loop of thread fixed with a pin so the pen will rotate around the centre
The length of the thread will determine the size of the circle. Alternately filled grids arising from concentric circles against a convergent axis lead to a flat surface but when stared at, these perceptually oscillate, circles advancing and receeding while triangular sections reach into the centre and emerge from it.

'---/Ellipses are constructed (using two pins and a loop of thread). Rotating a pencil tied to a thread around two foci seeing that the point describing the line is controlled by both.---/'

Fig.79. The pen being constrained by both of the pins

Fig.80. These lines are used to construct a triple ellipse
Both of these foci function in relation to each other determining a whole complex which flips into opposite directions.

Fig.81. The alternately filled intervals arising from the figure, oscillate between flat patterns and separate levels.

Fig.82. Randomly filled  intervals arising from fig.81 retain the impression of a cellular or a  galactic unity attracted by a centre but floating around in a sort of gravitation.

.XI. JEWEL, S.W. Hayter
1986, Catalogue, Galerie J.C.Riedel 12 Rue Guénégaud, Paris
Here the specific 'Jewel' section includes the flat patterns and the moving effects from figures 80,81 and 82.

The exposition continues with a simplified description of how to construct parabolas from straight lines.
'---/Parabolas (marking out regular intervals on two asymptotes joining 10:1, 9:2, 8:3 etc), hyperbolas, leaving all construction lines as well as the form sought for. Two parabolas having one asymptote in common can be used to construct a vortex. Four can be used to give an illusion of sculptural form.'
Parabolas and vortices:The asymptote is by definition nearer and nearer to a given curve but does not reach it within a finite distance so that Hayter's process, while being illuminating, is strictly speaking inaccurate. Parabolas and vortices are constructed one out of the other. The parabola is constructed from the same lines that would be used for a vortex but emphasing the round section rather than the concave.

.XII. VORTEX, S.W. Hayter
Fig 105, page 225, Chapter 17, 'New Ways of Gravure', 1981, New York, Watson-Guptill Publications
These two constructed parabolas never quite reaching their asymptote but approaching it continuously reveal a vortex opening in the distance between them.

.XIII. VORTEX , S.W. Hayter
fig 105, page 225, Chapter 17, 'New Ways of Gravure', 1981, New York, Watson-Guptill Publications
The vortex's rotary motion is emphasized by the encircling filled grids that surround it and give it form.

Fig.83. Two vortices joined at their widest yield a three-dimensional figure which rises forward toward their centre.

Fig.84. The same figure with alternately filled intervals emphasizes this rising  through the distorted shapes of the rectangles.

Fig.85. Construction lines creating at their limits two hyperbolas. These straight lines seem to bulge outward toward their centre.

Fig.86. The construction lines having been rotated through 90° and filled in alternately, possess a convicing solidity, bulging outward and contracting inward,oscillating continuously but they are actually only hyperbola constructing lines.

These can be further deformed by placing them within spaces of curved parallels.

Fig.87. Two vortices joigned at their widest sections against a background of concave curves which fit into them.

Fig.88. Two vortices joined at their widest sections are projected against a background of vertical parallel lines which push them forward and outward.

Fig.89. Construction lines of two parabolas joined at right angles are hovering against a background of concave curves which tend to create a counter-clockwise movement.

Fig.90. These construction lines are displayed against a background of vertical parallel lines which push them forward.

Fig.91. Alternately filled intervals arising from concentric ellipses against a background of concave curves vary between presenting a two dimensional grid to a three dimensional complexity of these ellipses. Expanding circles on the surface of the water produced concretly by thrown stones create a spontaneous moiré pattern.

Fig.92. The construction lines of three ellipses are drawn without the ellipses themselves being present. The construction lines oscillate dynamically, going forward and then retreating and are elevated by the rings of concentric circles.

There is no end to such construction of figures.

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