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This
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page
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describes
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the
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formulae
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used
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project
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tracking
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errors
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to
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the
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ACD
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These
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are
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written
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up
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in
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greater
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detail
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(and
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with
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real
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formulae,
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instead
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of
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text-based
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approximations)
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here:
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General framework
The tracker has a 4x4 representation (x_tkr) which uses a pair of slope-intercepts (x_0,y_0,m_x,
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m_y)
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to
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parametrize
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the
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position
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as
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a
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function
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of
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z.
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This
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is
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great
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for
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a
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bunch
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of
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planes
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normal
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to
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the
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z
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axis,
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but
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really
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not
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so
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nice
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for
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planes
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normal
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to
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the
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x
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and
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y
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axes.
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The
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first
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order
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of
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business
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is
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to
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transform
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the
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track
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parameterization
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to
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a
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5x5
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representation
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(x_acd
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)which
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uses
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a
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Point
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at
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Z=0
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and
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a
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Vector
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(x_0,y_0,
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v_x,v_y,v_z)
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to
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parameterize
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the
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positions
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by
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arclength
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along
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the
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track
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(s).
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We
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can
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the
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transform
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the
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covarience
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matrix
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by
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using
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the
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5x4
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track
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representation
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derivative
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matrix
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A
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=
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d
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x_acd
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/
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d
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x_tkr:
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C_acd
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(5x5)
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=
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A
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C_tkr
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(4x4)
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A_T
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| Wiki Markup |
|---|
If we then use the track parameterization to calculate a \[point of closest approach (POCA) or an intersection point |
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(p |
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), the covarience on that point can be found using 5x3 solution derivative matrix |
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B = d p / d A_acd
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with
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the
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covarience
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given
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by:
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C_p
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(3x3)
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=
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B
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C_acd
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(5x5)
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B_T
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In
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the
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case
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where
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we
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care
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about
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the
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distance
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of
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closest
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approach
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to
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a
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point
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or
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a
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ray,
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we
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can
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project
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along
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the
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vector
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of
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closest
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approach
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J
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(the
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vector
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from
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the
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POCA
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to
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the
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point
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or
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ray
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in
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question).
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Then we find the error on the distance of closest approach:
C_doca (1x1) = J C_p(3x3)
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J_T
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In
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the
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case
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where
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we
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care
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about
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the
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intersection
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of
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a
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track
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with
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a
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plane,
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we
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can
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rotate
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into
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the
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plane
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using
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the
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rotation
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matrix
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R.
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C_plane
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(3x3)
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=
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R
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C_p(3x3)
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R_T
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and
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take
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only
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the
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upper-left
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4
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entries
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to
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get
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the
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2x2
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error
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ellipse
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on
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the
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surface
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of
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the
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plane.
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Errors on POCA between a track and a point
Call the point q, and the track (x,v)
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the
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POCA
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occurs
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at
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an arclength
s = (x - q) . v
therefore the poca is
p = x + sv
and the vector of closest approach is:
J = q-p = q - x + sv
The full derivative matrix (B) is given in the attached write up.
Errors on POCA between a track and a ray
| Wiki Markup |
|---|
Call the ray (r,q) and the track (x,v), the POCA occurs at an arclength along the track
s = \[b(r.(x-q) - v.(x-q)\] / \[1-b*b\]
and an arclength along the ray:
t = \[r.(x-q) - b(v.(x-q))\] / \[1-b*b\]
where b = v.r |
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Then
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the
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poca
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and
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vector
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of
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closest
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approach
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are:
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p = x + sv
J = (x-q)
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+
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sv
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-
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tr
The full derivative matrix (B) is given in the attached write up.
Errors on point where a track crosses a plane
| Wiki Markup |
|---|
Given a plane defined by a point q and a rotation matrix R, the intersection occurs at:
s = \[ (x-q).R_z \] / \[ v. R_z \] |
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Then
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the
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intersection
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point
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in
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the
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global
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frame
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is:
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p = x + sv
Of course we can get the intersection point in the local frame l:
l = R p + q
The full derivative matrix (B) is given in the attached write up.
Some issues with the Error Projection
- They depend on the Kalman fitter hypothesis. Describing a 8GeV proton as a 100MeV electron (or vice-versa) is not going to give accurate errors
- They rely on knowing the material along the track path. For low energy particle the errors can blow up very quickly if there is a lot of material between the head of the track and the ACD
We distinguish between to ways of calculating the error projections:
- Full Propagation: track is kalman-propagated all the way back to the POCA and errors are calculated from the cov. matrix at the POCA
- Projection: errors are taken at the head of the track projected out at the POCA.
The errors from method 1) will alway be larger than the error from 2). For low-energy tracks the difference can be quite large.