- WELCOME TO AN INTRODUCTION TO CONIC SECTIONS.
THIS IS THE FIRST OF SEVERAL VIDEOS
THAT WILL REVIEW THE CONIC SECTIONS.
LET'S FIRST START OFF BY TALKING ABOUT SOME VOCABULARY.
IF WE START WITH A DOUBLE CONE, AS WE SEE HERE IN GREEN,
THE FIXED LINE IS CALLED THE AXIS OF THE CONE.
AND THAT WOULD BE THIS LINE HERE.
NEXT THE VERTEX IS THE POINT SHARED BY BOTH CONES
AND THAT POINT WOULD BE RIGHT HERE.
NEXT THE LINES THAT PASS THROUGH THE VERTEX
AND FORM THE CONES ARE CALLED THE GENERATORS.
SO THE GENERATORS LIE IN THE CONE.
AND THERE WOULD BE AN INFINITE NUMBER OF THEM
THAT FORM THE TWO CONES.
AND THEN LASTLY THE CONE CONSISTS OF TWO PARTS
CALLED THE NAPES.
SO THE WAY WE CREATE THE CONIC SECTIONS IS TO TAKE A PLANE
AND INTERSECT THIS DOUBLE CONE.
AND BASED UPON HOW THE DOUBLE CONE IS SLICED
OR INTERSECTED IT EFFECTS WHAT CONIC SECTION IS FORMED.
LET'S TAKE A LOOK AT THE FOUR NONE DEGENERATE CONIC SECTIONS.
THE FIRST IS THE CIRCLE, AS WE SEE HERE IN RED.
THE NEXT IS THE ELLIPSE WHICH LOOKS MORE
LIKE A FLATTENED CIRCLE.
IN BLUE WE HAVE THE PARABOLA
WHICH YOU SHOULD ALREADY BE FAMILIAR WITH.
AND THEN LASTLY WE HAVE THE HYPERBOLA.
AND THIS SHAPE IS DIFFERENT THAN THE PARABOLA.
LET'S TAKE A LOOK AT HOW THESE FOUR CONICS ARE FORMED.
THE CIRCLE AS WE SEE HERE IS GENERATED
WHEN THE PLANE IS PERPENDICULAR TO THE AXIS OF THE CONE.
REMEMBER THE AXIS WOULD BE THIS FIXED LINE AS WE SEE HERE.
SO IF THE PLANE IS PERPENDICULAR TO THIS LINE WE HAVE A CIRCLE.
THE ELLIPSE, AS WE SEE HERE ABOVE THE CIRCLE,
IS GENERATED WHEN THE PLANE IS TILTED SO IT INTERSECTS
EACH GENERATOR BUT ONLY INTERSECTS ONE OF THE NAPES.
ON THE LEFT WE SEE THE PARABOLA AND THIS IS GENERATED
WHEN THE PLANE IS TILTED SO IT IS PARALLEL TO ONE GENERATOR
AND ONLY INTERSECTS ONE OF THE NAPES.
WE COULD ALSO SAY THAT IT'S PARALLEL
TO ONE SIDE OF THE CONE.
AND THEN LASTLY, ON THE RIGHT THE HYPERBOLA IS GENERATED
WHEN THE PLANE INTERSECTS BOTH NAPES.
LET'S TAKE A LOOK AT THIS ANOTHER WAY.
HERE WE HAVE A DOUBLE CONE IN THIS LIGHT BLUE PLANE
INTERSECTING THE CONE.
IF YOU CAN IMAGINE THE ACCESS PASSING THROUGH THESE TWO CONES
THIS PLANE IS PERPENDICULAR TO THAT LINE.
IF WE TAKE A LOOK AT THE CONIC SECTION
THAT'S GENERATED YOU CAN SEE THAT IT'S A CIRCLE.
NOW, AS WE CHANGE THE ANGLE OF INCLINATION OF THAT PLANE
IT'S NO LONGER PERPENDICULAR TO THE ACCESS
BUT IT'S STILL INTERSECTING JUST ONE NAPE.
IN THIS CONIC SECTION WOULD BE THE ELLIPSE AS WE SEE HERE.
IF WE CHANGE THE ANGLE OF INCLINATION,
AS LONG AS IT ONLY INTERSECTS ONE NAPE,
IT IS STILL AN ELLIPSE.
NOW, IF WE CHANGE THE ANGLE OF INCLINE
SO THAT IT'S PARALLEL TO ONE OF THE GENERATORS...
AS WE SEE HERE THE CONIC SECTION WOULD BE A PARABOLA.
AND THE HYPERBOLA IS GENERATED
WHEN THE PLANE INTERSECTS BOTH NAPES.
SO THIS WOULD BE THE HYPERBOLA AND WE CAN SEE THE SHAPE...
AND THERE YOU CAN SEE THE SHAPE OF A HYPERBOLA.
OKAY, SO THAT PRETTY MUCH DOES IT FOR THIS FIRST VIDEO
ON THE INTRODUCTION TO CONICS.
WE'VE DESCRIBED THE FOUR NON-DEGENERATE CONIC SECTIONS
AND WE'VE ALSO SEEN HOW THEY CAN BE GENERATED
USING A PLANE AND A DOUBLE CONE.
THE NEXT SERIES OF VIDEOS
WE'LL TAKE A LOOK AT EACH CONIC SECTION INDIVIDUALLY
AND WE'LL ALSO GRAPH THEM IN STANDARD AND GENERAL FORM.
SO THANK YOU FOR WATCHING.