English language

How to pronounce quadric surface in English?

Full Transcript

let's do a sketch of a quadric surface
this quadric surface is going to be
defined by the equation
x squared over 9 plus y squared
over 16 plus z
squared over 25 equals
one
one of the first things you should know
about this quadric surface is one
it's a quadrant surface what that means
is
the highest degree polynomial between
x y and z is to the second degree
since it's to the second degree if this
were in let's say
two dimensions you would say hey that's
a quadratic equation
if it was a two dimensional function of
one
independent variable here however
we have three variables x y and z
but none of the variables are higher
than the second order
since they're not higher than the second
order then
and the highest degree is a second order
degree
term then we would consider this a
quadric surface
now we've reviewed quadric surfaces in
another video so take a look if you
don't remember what they are
the thing about this quadric surface is
we
recognize this one to be the quadric
surface
for an ellipse
now you may not recognize that right
away with practice you will
if we were to try and sketch the c-lips
one thing that would help us
dramatically in even
recognizing that the surface is an
ellipse
is to sketch the traces of the surface
now remember a trace is the line of
intersection
between a plane that is parallel to
one of the coordinate planes
and intersects a surface
so for this let's construct three traces
let's construct a y z trace this will
give us a
line of intersection in the y z plane
we'll construct an
x y trace which will give us a line of
intersection in the
x y plane and we'll construct a z x
trace which will give us a line of
intersection in the z x plane
with these three traces we'll be able to
get a good sense for what this ellipse
looks like
let's begin by sketching our x y z
rectangular coordinate system and in
fact just so we can be
a little bit more accurate let's go
ahead and
do it on some graph paper the vertical
axis will correspond to the
z-axis the horizontal axis will
correspond to the y-axis and the axis
coming
straight out of the screen will
correspond to the x-axis
let's do a nice consistent scaling on
each of the axes
every one of these divisions will be
separated by the next division
by one unit along the y-axis
also by one unit along the axis
since the x-axis is coming straight at
us
out of the page and since it's skewed a
little bit
that one is going to be harder to get a
nice consistent scaling so
i'll just guesstimate something that
looks
roughly like this
hopefully that'll be close enough
let's go ahead and determine the x y
and z intercepts of
our ellipse so to begin with the
x-intercept
we know that the x-intercept
is going to be found by setting the
variables
y and z and z equal to zero
when we set those variables equal to
zero we get
the equation x squared over
nine is equal to one solving for x we
get
x equals plus or minus
three
let's put that on our coordinate system
here is x equals three and here is x
equals minus three
similarly for the y-intercept we're
going to set
x equals zero and z equals zero
in doing that we get the equation y
squared
over 16 is equal to one
solving for y we get y equals
plus or minus four
here's y equals plus four here is y
equals minus four
for the z-intercept it's more of the
same
except this time we'll set x equals to
zero
and y equal to zero this gives us the
equation
the z squared over 25
is equal to one solving for z
we get z is equal to plus or minus
5.
so now we have the intercepts
of what we are thinking is going to be
an ellipse because of the nature of the
quadratic
equation the quadric surface
if i were to guess i would guess that
the ellipse
has or i i said ellipse actually
this is an ellipsoid in
three-dimensional space
it would be an ellipse in
two-dimensional space
but an ellipsoid is what we call it in
three dimensional space
well now that we have the intercepts of
the ellipsoid
let's see if we can sketch it now to
sketch it we're going to sketch it by
beginning
with the various traces
now as we've said earlier a trace
is the line of intersection between
a plane that is parallel to one of the
coordinate planes
and the surface in question
now that line of intersection will often
produce a recognizable shape
if the if if the intersection between
the plane
and a quadric surface occurs
then the line of intersection will
typically be
one of the conic sections that we're
familiar with
from previous from from previous
semester
well let's begin by doing a y z
trace
so to do a y z trace
let's choose the plane that is parallel
to the y
z axis
so i am going to do a plane if it's
parallel
to the y z axis then a normal
vector would be a vector that points in
the
positive x direction
well let's do if we're doing the y
z trace then that is going to be the
plane
with the equation x equals zero
setting x equals to zero in our equation
gives us y squared over 16
plus z squared over 25
is equal to one
notice that the yz trace is
the equation of an ellipse
and this is an ellipse in the y z
plane being an ellipse
we have a major axis of
5 a total length of 10 on the major axis
and a minor axis of 4 or a radius on the
minor axis of 4
along the y-axis if we were to sketch
this ellipse
in the yz plane it would look similar to
this
and this corresponds to our yz
trace
let's do another trace
let's do the x y trace
in doing the x y trace we would do
a plane that is parallel to the x
y coordinate plane so that plane
parallel to the x y
coordinate plane is going to have a
vector perpendicular to it
that is parallel to the z axis
so the plane for a
the equation of a plane in the x
y plane is simply z equals zero
setting z equals to zero in our
quadratic equation
gives us x squared over nine plus
y squared over sixteen
is equal to one
well notice this is just an equation of
another ellipse
except this ellipse is in the x y plane
the major axis has a length of
eight and the minor axis has a length
of six
sketching this in the x y plane
would produce an ellipse that looks
something like this
and notice that's now looking a lot
like an ellipsoid an
egg if you will
with this i think we're pretty convinced
that we have an ellipsoid
however let's just do one more trace
the last trace is going to be
the z x
trace
the zx trace is obtained
by taking a plane that is
parallel to the zx plane
that plane that is parallel to the zx
plane
will have a unit vector that is
parallel to the y-axis and perpendicular
to the z-x plane
so our z-x trace is going to have the
value
y equals 0 describing the plane
well that's not the trace that's the
equation for the plane
so now plugging that into our quadratic
equation
we get an equation that is x squared
over nine
plus z squared over twenty five
is equal to one we have
a third ellipse
this ellipse is in the zx plane
adding that to our sketch gives us a
little bit more perspective
that ellipse in the zx plane looks
something like
this
and now we've completed our sketch
of an ellipsoid given by the quadratic
surface x squared over 9 plus y squared
over 16
plus z squared over 25 is equal to 1.
and i want you to notice how our
ellipses
in each of the coordinate planes
for this example happens to pass through
the intercepts
i don't think we really need it to plot
the intercepts
in this in this example other than to
maybe it helps fixes
fix our eyes onto where
the ellipse is and its shape and
perspective
but in general whether i think i need an
intercept or not i'll typically plot it
if one exists to give me more
information
and at a minimum i'll do
a trace at least two traces
parallel to two different coordinate
axes
for now we've completed our sketch well
done

let's do a sketch of a quadric surface

Toggle Transcript

quadric surface::noun | Spocien Dictionary

quadric surface

noun

/'kwɑdrɪk_'sɜrfɪs/

a curve or surface whose equation (in Cartesian coordinates) is of the second degree

Type	Words
Synonyms	quadric
Type of	curve, curved shape
Has types	hyperboloid

Examples of quadric surface

quadric surface

An ellipsoid is a closed quadric surface that is a three dimensional analogue of an ellipse.

From the en.wikipedia.org

In mathematics, a paraboloid is a quadric surface of special kind.

From the en.wikipedia.org

How to pronounce quadric surface in English?

quadric surface::noun | Spocien Dictionary

abandonment

Related words

Examples of quadric surface

Related

Subscribe to our Newslatter