Envelope (mathematics)

In geometry, an envelope of a family of curves in the plane is a curve that is tangent to each member of the family at some point. Classically, a point on the envelope can be thought of as the intersection of two "adjacent" curves, meaning the limit of intersections of nearby curves. This idea can be generalized to an envelope of surfaces in space, and so on to higher dimensions.

Envelope of a family of curves

Let each curve C_t in the family be given as the solution of an equation f_t(x, y)=0 (see implicit curve), where t is a parameter. Write F(t, x, y)=f_t(x, y) and assume F is differentiable.

The envelope of the family C_t is then defined as the set of points for which

for some value of t, where is the partial derivative of F with respect to t.

Note that if t and u, t≠u are two values of the parameter then the intersection of the curves C_t and C_u is given by

or equivalently

Letting u→t gives the definition above.

An important special case is when F(t, x, y) is a polynomial in t. This includes, by clearing denominators, the case where F(t, x, y) is a rational function in t. In this case, the definition amounts to t being a double root of F(t, x, y), so the equation of the envelope can be found by setting the discriminant of F to 0.

Synthesizer

A sound synthesizer (usually abbreviated as "synthesizer" or "synth", also spelled "synthesiser") is an electronic musical instrument that generates electric signals that are converted to sound through instrument amplifiers and loudspeakers or headphones. Synthesizers may either imitate the still existing sounds (instruments, vocal, natural sound, etc.), or generate new electronic timbres not existing before. They are often played with a musical keyboard, but they can be controlled via a variety of other input devices, including music sequencers, instrument controllers, fingerboards, guitar synthesizers, wind controllers, and electronic drums. Synthesizers without built-in controllers are often called sound modules, and are controlled via MIDI or CV/Gate using a controller device.

Synthesizers use various methods to generate signal. Among the most popular waveform synthesis techniques are subtractive synthesis, additive synthesis, wavetable synthesis, frequency modulation synthesis, phase distortion synthesis, physical modeling synthesis and sample-based synthesis. Other less common synthesis types (see #Types of synthesis) include subharmonic synthesis, a form of additive synthesis via subharmonics (used by mixture trautonium), and granular synthesis, sample-based synthesis based on grains of sound, generally resulting in soundscapes or clouds.

Envelope detector

An envelope detector is an electronic circuit that takes a high-frequency signal as input and provides an output which is the envelope of the original signal. The capacitor in the circuit stores up charge on the rising edge, and releases it slowly through the resistor when the signal falls. The diode in series rectifies the incoming signal, allowing current flow only when the positive input terminal is at a higher potential than the negative input terminal.

Most practical envelope detectors use either half-wave or full-wave rectification of the signal to convert the AC audio input into a pulsed DC signal. Filtering is then used to smooth the final result. This filtering is rarely perfect and some "ripple" is likely to remain on the envelope follower output, particularly for low frequency inputs such as notes from a bass guitar. More filtering gives a smoother result, but decreases the responsiveness; thus, real-world designs must be optimized for the application.

Definition of the envelope

Sunrise (1926 film)

Sunrise is a 1926 Australian silent film co-directed by Raymond Longford, who took over during filming. It was the second film from Australasian Films following their recommencement of production, after Painted Daughters.

It is considered a lost film.

Plot

George Willis loses his unfaithful wife in a rock fall and takes to the bush He rescues a girl, Hope Stuart, from a flood and nurses her back to health. When he brings her back to her father he discovers that an old enemy, Arthur Greerson, has accused him of murder.

Greerson is injured in a mining accident and after George rescues him, Greerson admits he has lied. George returns to his life as a recluse in the mountains, followed by Hope.

Cast

Production

The film was shot on location in the Blue Mountains at Bargo with interiors at the studios of Australasian Films in Bondi. F. Stuart Whyte began directing but left Australia during shooting for unknown reasons. He was replaced by Longford, who had recently contracted to Australasian Films.

Sunrise (TV program)

Sunrise is an Australian breakfast television program, broadcast on the Seven Network and currently hosted by David Koch and Samantha Armytage. The program follows Seven Early News and runs from 5:30 am to 9:15 am, followed by The Morning Show.

History

The history of Sunrise can be traced back to at least 17 January 1991 when 11AM news presenter Darren McDonald began presenting an early morning Seven News – Sunrise Edition bulletin prior to hostilities breaking out during the Gulf War.

In 1996, Seven introduced a one-hour weekday bulletin called Sunrise News, later renamed Sunrise. Seven recruited Chris Bath from NBN Television to present the bulletin alongside Peter Ford. Ford moved to other presenting roles in 1996 and was replaced by finance editor David Koch. In 1997, Chris Bath was transferred to Seven's 10.30 pm News and was replaced by Melissa Doyle. In 1998 Sunrise was presented by Doyle and Nick McArdle. Seven launched a Sunday bulletin hosted by Stan Grant, entitled Sunday Sunrise, in 1997. Weekday Sunrise was cancelled in 1999, replaced by children's program The Big Breakfast. Seven maintained half-hourly news updates during The Big Breakfast and their Sunday bulletin was not affected by the axing. Other temporary Sunrise hosts up until this time include Anne Fulwood (who was filling in for Georgie Gardner at the time of the program's end), Leigh Hatcher and Nick McArdle (previously weekend sport presenter, Seven News Sydney).

Sunrise (band)

Sunrise is a German pop in the late 1970s. In 1978, the song Call On Me was covered in German by Jürgen Drews.

The band's greatest hits include Call On Me (1977) and Good, good, good (1979). Their Album Call On Me was produced by the German musicians Holger Julian Copp and Hanno Harders (also known as Beagle Music Ltd.) and Michael Reinecke.

References