1. The document provides instructions and examples for solving problems involving midpoint and distance formulas, as well as transformations in the coordinate plane. Practice problems are given for finding midpoints and distances between points, as well as homework instructions.
2. Key terms related to transformations are defined, including reflection, rotation, translation, arrow notation, and the rule for describing translations.
3. Examples are given of transformations, including reflections across lines and rotations about points. Formulas and notation for describing translations are also provided.
The midpoint method or technique is a “measurement” and as each measurement it has a tolerance, but
worst of all it can be invalid, called Out-of-Control or OoC. The core of all midpoint methods is the accurate
measurement of the difference of the squared distances of two points to the “polar” of their midpoint
with respect to the conic. When this measurement is valid, it also measures the difference of the squared
distances of these points to the conic, although it may be inaccurate, called Out-of-Accuracy or OoA. The
primary condition is the necessary and sufficient condition that a measurement is valid. It is comletely
new and it can be checked ultra fast and before the actual measurement starts. .
Modeling an incremental algorithm, shows that the curve must be subdivided into “piecewise monotonic”
sections, the start point must be optimal, and it explains that the 2D-incremental method can find, locally,
the global Least Square Distance. Locally means that there are at most three candidate points for a given
monotonic direction; therefore the 2D-midpoint method has, locally, at most three measurements.
When all the possible measurements are invalid, the midpoint method cannot be applied, and in that case
the ultra fast “OoC-rule” selects the candidate point. This guarantees, for the first time, a 100% stable,
ultra-fast, berserkless midpoint algorithm, which can be easily transformed to hardware. The new algorithm
is on average (26.5±5)% faster than Mathematica, using the same resolution and tested using 42
different conics. Both programs are completely written in Mathematica and only ContourPlot[] has been
replaced with a module to generate the grid-points, drawn with Mathematica’s
Graphics[Line{gridpoints}] function.
We define the definite integral as a limit of Riemann sums, compute some approximations, then investigate the basic additive and comparative properties
The midpoint method or technique is a “measurement” and as each measurement it has a tolerance, but
worst of all it can be invalid, called Out-of-Control or OoC. The core of all midpoint methods is the accurate
measurement of the difference of the squared distances of two points to the “polar” of their midpoint
with respect to the conic. When this measurement is valid, it also measures the difference of the squared
distances of these points to the conic, although it may be inaccurate, called Out-of-Accuracy or OoA. The
primary condition is the necessary and sufficient condition that a measurement is valid. It is comletely
new and it can be checked ultra fast and before the actual measurement starts. .
Modeling an incremental algorithm, shows that the curve must be subdivided into “piecewise monotonic”
sections, the start point must be optimal, and it explains that the 2D-incremental method can find, locally,
the global Least Square Distance. Locally means that there are at most three candidate points for a given
monotonic direction; therefore the 2D-midpoint method has, locally, at most three measurements.
When all the possible measurements are invalid, the midpoint method cannot be applied, and in that case
the ultra fast “OoC-rule” selects the candidate point. This guarantees, for the first time, a 100% stable,
ultra-fast, berserkless midpoint algorithm, which can be easily transformed to hardware. The new algorithm
is on average (26.5±5)% faster than Mathematica, using the same resolution and tested using 42
different conics. Both programs are completely written in Mathematica and only ContourPlot[] has been
replaced with a module to generate the grid-points, drawn with Mathematica’s
Graphics[Line{gridpoints}] function.
We define the definite integral as a limit of Riemann sums, compute some approximations, then investigate the basic additive and comparative properties
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http://www.slideshare.net/JamesSmith245/solution-of-the-ccp-case-of-the-problem-of-apollonius-via-geometric-clifford-algebra
http://www.slideshare.net/JamesSmith245/rotations-of-vectors-via-geometric-algebra-explanation-and-usage-in-solving-classic-geometric-construction-problems-version-of-11-february-2016
http://www.slideshare.net/JamesSmith245/resoluciones-de-problemas-de-construccin-geomtricos-por-medio-de-la-geometra-clsica-y-el-lgebra-geomtrica-vectorial
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All of this illustrated with link prediction over knowledge graphs, but the argument is general.
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Connector Corner: Automate dynamic content and events by pushing a buttonDianaGray10
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Send an interactive Slack channel message (using buttons)
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In a second workflow supporting the same use case, you’ll see:
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If the “Approve” button is clicked, a Jira/Zendesk ticket is created for the marketing design team
But—if the “Reject” button is pushed, colleagues will be alerted via Slack message
Join us to learn more about this new, human-in-the-loop capability, brought to you by Integration Service connectors.
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Akshay Agnihotri, Product Manager
Charlie Greenberg, Host
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Sidekick Solutions uses Bonterra Impact Management (fka Social Solutions Apricot) and automation solutions to integrate data for business workflows.
We believe integration and automation are essential to user experience and the promise of efficient work through technology. Automation is the critical ingredient to realizing that full vision. We develop integration products and services for Bonterra Case Management software to support the deployment of automations for a variety of use cases.
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1. Opener:
1. Find the coordinates of the midpoint of PQ with endpoints P(‐8, 3) and Q(‐2, 7).
2. Find XM. X has coordinates (2, 7), and M has coordinates (6, 1).
1
2. Extra Distance & Midpoint Practice:
1. If CD = 12 and C(4, ‐7) and D(2, y), what's the value of y?
2. If the midpoint of a segment is (8, ‐2) and one of the endpoints is (12, 6), what
is the other endpoint?
2
3. Homework Add‐On:
Choose at least 3 of the problems that you got deductions on from
your quiz.
Write a new quiz problem that tests the same skill in a similar way
(using a quadratic for example in #15).
Solve each problem, showing your work to demonstrate that you
understand your mistakes.
If you do more than 3, you may receive some type of reward for your
efforts...
3
4. Section 1.7: Transformations in the Coordinate Plane
Some vocabulary:
1. Transformation: a change in the position,
size or shape of a figure
2. Preimage: the original figure
3. Image: the resulting figure after
the transformation
4. Reflection: (flip) a transformation
across a line
5. Rotation: (turn) a transformation
about a point
6. Translation: (slide) a transformation where
all points of a figure move the same
distance in the same direction
7. Arrow Notation: a way of describing transformations
ABC ‐‐> A'B'C'
8. Rule: descriptors for translations ‐ (x, y) ‐‐> (x + a, y + b)
4
