Warning, danger, danger... Proceed at substantial risk! Life threatening electrical levels are used in this project which can present the danger of electrocution.... If you have no experience in electrical wiring find someone who does to assist you! (or drop this project!)
Also, hot wire cutting of many plastics releases noxious and in some cases deadly fumes... use outside or with more than adequate ventilation. The wire and resistor get hot enough to burn you!
Still sure you want to proceed AT YOUR OWN RISK? You can cut foam with scissors you know... just don't run with them.The vital calculation!
You must limit the amperage to the Ni-chrome wire with the wire wound resistor or you run the risk of blowing the house fuse. (or circuit breaker) You may also ruin your transformer with excessive current demands. Resistor means it resists current flow which generates heat. Both the Ni-chrome wire and the resistor will get hot.
The trick is to get enough amperage to make your Ni-chrome wire hot without damaging any of the components. (You don't need to get it glowing red like a toaster!) I suggest 2-3 amps as sufficient.
So... you should use ohms law to calculate amperage like this:
My resistance for the Ni-chrome wire is 1.3 ohms per foot. (yours may vary)
My wire wound resister total resistance is 6 ohms. (using one side, mine has two windings, 6 and 2 ohms, yours may be different)
My transformer (as measured by a multimeter) output is 17.8 volts. (surprised me, the surplus store item was advertised as a 12 volt transformer!)
Amperage = volts divided by total resistance...
3. ohms + 6 ohms = 7.3 ohms divided into 17.8 volts = 2.438 amperes current available.
(Ni-chrome) + (resistor) = 7.3 ohms
Items you may need:

A) Soldering gun (or iron)
B) A multimeter
C) Household electric plug with attached wire
D) Transformer, any rated at 3 amps or more delivering from 5-12 volts
E) Ni-chrome wire (Nickel & chrome composition which heats up red hot without breaking)
F) Rosin core Solder
G) Connecting wire
H) Electrical connectors
I) Ceramic case wire wound resistor
J) Wire stripper tool
Sources of supply: Surplus or retail electronics stores, hardware stores, internet venders, etc.
Yet another warning for the truly dim...
This circuit has no on/off switch. I you leave it plugged in and unattended you may burn your house down. If you do you cannot move in with me.Connecting it all together... the 'bread-boarded' circuit with due mischief done to a styro egg.

The power cord is connected to the primary coil of the transformer. One of the transformer secondary outputs (the low voltage side) is wired/soldered to the wire-wound resistor. The other end of the resistor is soldered and wired directly to the Ni-chrome wire. The other transformer secondary output is wired to the other end of the Ni-chrome completing this simple circuit. See the metal plate beneath the resistor? That is to help dissipate the heat so it does not burn my pristine workbench top. Also, note that I have simply stretched the Ni-chrome between two nails for this 'smoke' test... If all goes well up to this point you are ready for the next step, housing the circuit and fabrication of a 'stretcher/holder'.
Housing the circuit... I advise metal to dissipate heat. Do make sure good wiring practices are followed so that any 'hot' wires are insulated and kept away from the housing! This is a coffee can which works just fine. The object is to keep any potentially harmful voltages well away from the potential users. Two longer wire leads were added from the transformer and resister so as to allow remote connection to a handle of sorts.


As to the 'stretcher/holder' thingee... hopefully this picture of a quite simple contrivance is worth a thousand words. The object is to stretch the Ni-chrome wire between two points maintaining suitable space and electrical separation between those points.
Many design modifications could be used... you might for instance prefer to stretch your wire vertically with a different type of structure simulating a band-saw.
Work safe and best of good Luck! - rocky