A triangle wave is a non-sinusoidal
waveform named for its triangular
shape. It is a periodic, piece wise
linear, continuous real function;
|
Like a square wave, the triangle wave
contains only odd harmonics;
however, the higher harmonics roll off
much faster than in a square wave
(proportional to the inverse square
of the harmonic number as opposed
to just the inverse): a square wave
is a non-sinusoidal periodic waveform
in which the amplitude alternates at a steady
frequency between fixed minimum
& maximum values, with the same
duration at minimum and maximum;
Although not realizable in physical
systems, the transition between
minimum and maximum is
instantaneous for an ideal square wave;
Non-sinusoidal wave forms are
wave forms that are not pure sine waves:
They are derived from simple
math functions. While a pure sine
consists of a single frequency,
non-sinusoidal wave forms can be
described as containing multiple sine In knot theory,
a Lissajous knot is a knot defined
by parametric equations
of the form
{\displaystyle x=\cos(n{x}t+\phi _{x}),
\qquad y=\cos(n{y}t+\phi {y}),
\qquad z=\cos(n{z}t+\phi {z}),} x=\cos(n{x}t+\phi {x}),\qquad y=\cos(n{y}t+\phi {y}),\qquad z=\cos(n{z}t+\phi {z}),
A Lissajous 821 knot
where {\displaystyle n{x}} n{x},
{\displaystyle n{y}} n{y}, & {\displaystyle n{z}}
n{z} are integers and the phase shifts
{\displaystyle \phi _{x}} \phi _{x},
{\displaystyle \phi _{y}} \phi _{y}, &
{\displaystyle \phi _{z}} \phi _{z} may be any real numbers. [1] . . .
waves of different frequencies.
These "component" sine waves
will be whole
number multiples of a fundamental
or "lowest" frequency. The frequency
& amplitude of each component
can be found using a mathematical
technique known as Fourier analysis . . .
A Lissajous figure, made by releasing sand
from a container at the end a double pendulum
In mathematics, a Lissajous curve /ˈlɪsəʒuː/,
also known as Lissajous figure or Bowditch curve /ˈbaʊdɪtʃ/,
is the graph of a system of parametric equations
{\displaystyle x=A\sin(at+\delta ),\quad y=B\sin(bt),} x=A\sin(at+\delta ),\quad y=B\sin(bt),
which describe complex harmonic motion.
[This family of curves was investigated
by Nathaniel Bowditch in 1815,
and later in more detail by Jules Antoine Lissajous in 1857.]
The appearance of the figure is highly sensitive to the ratio
a
/
b
. For a ratio of 1, the figure is an ellipse,
with special cases including circles (A = B, δ =
π
/
2
radians) and lines (δ = 0). Another simple
Lissajous figure is the parabola (
b
/
a
= 2, δ =
π
/
4
).
Other ratios produce more complicated curves,
which are closed only if
a
/
b
is rational. The visual form of these curves
is often suggestive of a three-dimensional knot,
& indeed many kinds of knots, including those
known as Lissajous knots,
project to the plane as [Lissajous figures]
Visually, the ratio
a
/
b
determines the number of "lobes" of the figure.
For example, a ratio of
3
/
1
or
1
/
3
produces a figure with three major lobes
_(see: image) Similarly, a ratio of
5
/
4
produces a figure with five horizontal lobes & four vertical lobes. Rational ratios produce closed (connected) or "still" figures,
while irrational ratios produce figures that appear to rotate:
The ratio
A
/
B
determines the relative width-to-height ratio of the curve.
For example, a
ratio of:
2
/
1
produces a figure that is twice as wide
as it is high. Finally, the value of δ determines
the apparent "rotation" angle of the figure,
viewed as if it were actually a three-dimensional curve;
For example, δ = 0 produces x and y components
that are exactly in phase,
so the resulting figure appears as an apparent
three-dimensional figure viewed from straight
on (0°). In contrast, any non-zero δ produces
a figure that appears to be rotated, either as a left–right or an up–down
rotation (depending on the ratio
a
/
b
).
Lissajous figure on an oscilloscope, displaying a 1:3 relationship between the frequencies of the vertical and horizontal sinusoidal inputs, respectively.
Lissajous figures where a = 1, b = N (N is a natural number) and
{\displaystyle \delta ={\frac {N-1}{N}}{\frac {\pi }{2}}}
{\displaystyle \delta ={\frac {N-1}{N}}{\frac {\pi }{2}}}
are Chebyshev polynomials of the first kind of degree N.
This property is exploited to produce a set of points,
called Padua points, at which a function
may be sampled in order to compute either
a bi-variate interpolation or quadrature
of the function over the domain [−1,1] × [−1,1]:
