3D cones gradient waveform generation

A method of calculating the three dimensional cone trajectory, comprising:

• performing a search for the three dimensional cone trajectory with a specified time duration by performing a plurality of cycles, comprising:
• performing a search for a border success by performing a plurality of cycles, comprising:
• selecting a point on the three dimensional cone trajectory after the curvature starting point;
• working backward from the curvature starting point to reduce twist;
• providing a failure value if it is determined that when the twist reaches zero it is not possible to return to the origin at a final velocity of zero while meeting system constraints;
• and
• providing a success value if it is determined that when the twist reaches zero it is possible to return to the origin at a final velocity of zero while meeting system constraints;
• and
• stopping when the generated three dimensional cone trajectory has reached the specified time duration.

A magnetic resonance imaging apparatus, comprising:

• a magnetic resonance imaging excitation and detection system;
• and
• a controller electrically connected to the magnetic resonance imaging excitation and detection system, comprising:
• a display;
• at least one processor;
• and
• computer readable media, comprising:
• a) computer readable coded for using sampling requirements to define a three dimensional cone trajectory differential equation;
• b) computer readable code for solving the three dimensional cone differential equation to obtain a three dimensional cone trajectory starting from a curvature starting point spaced apart from an origin;
• c) computer readable code for performing a search for a border success by performing a plurality of cycles, comprising:
• computer readable code for selecting a point on the three dimensional cone trajectory after the curvature starting point;
• computer readable code for working backward from the curvature starting point to reduce twist;
• computer readable code for providing a failure value if it is determined that when the twist reaches zero it is not possible to return to the origin at a final velocity of zero while meeting system constraints;
• and
• computer readable code for providing a success value if it is determined that when the twist reaches zero it is possible to return to the origin at a final velocity of zero while meeting system constraints;
• d) computer readable code performing for a plurality of cycles, wherein each cycle comprises:
• applying a magnetic resonance imaging excitation;
• and
• scanning along the calculated three dimensional cone trajectory and acquiring a readout;
• e) computer readable code for producing a three dimensional image from the readout of the plurality of cycles;
• and
• f) computer readable code for displaying the three dimensional image on the display.